21 Electric Charge and Electric Field
Physics 7D
21 Electric Charge and Electric Field • •
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Electrostatics o interactions between electric charges that are at rest sign of charge related to attraction o two positively charge objects will repel same with negatively o a negative and a positive charge will always attract principle of electric charge o in a closed system, the sum of all electric chrages is constant an electron has the same charge magnitude as a proton o must be a whole number (can't have a fraction of a charge) transferring charges o conductor transfers charge has free electrons that are willing to travel o insulator does not transfer charge doesn't have many, or any, free electrons complicated molecules only a handful of elements (carbon, sulfur, phosphorus) are insulators rubbing 2 insulators together can result in one having a - charge while the other has a + o doesn't work for conductors induction o charging a body with the opposite sign of an object without losing the charge of the object o induced charge - areas of, say a ball, whose sides both have excess opposite charges o
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charged objects can attract an insulator
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Coulomb o the SI unit of electric charge o represents the magnitude of the charge of about 6*10^18 electrons an electron has a charge of 1.602*10^-19 Coulomb's law
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electric force between two point charges (N) = (constant)(|charge of one object * charge of the other object|)/(distance between them ^2) absolute values because the force can not be -, but the charges can constant usually = 9*10^9 (N m^2)/C^2 for 1 C charges. if 2 C, then multiply by 2 because epsilon-nought usually = 8.854*10^-12 C^2/(N m^2) o principle of superposition of forces Coulomb's law holds true for an infinite number of point charges ex. when two charges exert forces simultaneously on a third, the total force on that charge is the vector sum of the forces the two would exert individually force o equal and oppsoite (F[a on b] is - F[b on a])
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electric field o a body with a charge exerts an electric field whether another charge is there to feel it or not
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E is around A, test charge qo feels the force:
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F = ma = (charge)(electric field)
finding E
Electric Field Vector = (coonstant)(point charge)(unit vector between source point and field point; S and P)/(distance between the two)
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Vector field the varying from point to point of the electric field of a point charge for large objects, calculating the net electric force of point charges (infinite) on the body is the only way to accurately calculate the actual force
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Superposition of Electric Fields (non point charges)
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Electric Field Lines o demonstrates the direction of a system of electric fields o an individual electric field is tangent to any point on a line
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electric field lines bunched together means E is strong
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7 field lines can never intersect because their electric field has a unique direction field maps (cross sections of 3d patterns) toward -, away from + (duh)
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uniform electric field
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electric dipoles o pair of point charges with equal magnitudes and opposite signs (+/-) o example - water o
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the dipole makes it possible to lock molecules in place (Na+, Cl-) magnitude of dipole moment
magnitude of dipole moment (C*m) = (charge)(distance between them) the vector of p points toward the + o the net force on an electric dipole in a uniform external electric field always = 0 (not not the torque) o torque of a dipole while in an external electric field
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magnitude of torque = (magnitude of dipole moment)(magnitude of external electric field)(sin of the angle between p and E)
vector of torque on an electric dipole = cross product of p and E torque goes through the page (either out of in) and follows the right-hand rule in the direction of decreasing angle. the dipole (p) wants to be ||with E, AND pointing in the same direction in the example above, the torque is going into the page (rotating clockwise) o potential energy for a dipole in an electric field
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has a min when the angle = 0, pi, etc (||) has a max when the angle = pi/2, etc (+)
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22 Gauss's Law •
Flux o
the flow through an area
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where the box is imaginary
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calculating o V dot A where A is the area vector, V is the uniform velocity, or electric field, flowing through that area o
electric flux of a uniform, flat surface = electric field x area (cos(angle))
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flux is - if the A vector is choosen to point in the opposite direction of E 1D - linear charge denisty
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r^-2 2D - surface charge denisty
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= charge/area
r^-1 3D - volume charge denisty
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= charge/length
= charge/volume r^0 = 1
Gauss's Law o for a closed spherical surface o
electric flux = enclosed charge/constant independent of the radius o for any closed surface
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moral of the story: for any closed surface this in in theory. in practice, this is only true for symetircal surfaces o flux is - for a - inside charge (A is pointing the opposite way of E) o
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if the charge is outside the surface, the net flux is always 0 o
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where the chrages lie for solid conductors (metal ball) vs solid insulators (ball of yarn)
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conductor's inside - the net electric field inside a conductor is always zero. If the net electric field were not zero, a current would flow inside the conductor. This would build up charge on the exterior of the conductor. This charge would oppose the field, ultimately (in a few nanoseconds for a metal) canceling the field to zero.
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placing a charge inside a conductor o
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23 Electric Potential •
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electric potential o gravitational potential energy depends on the hight of a mass as electric potential depends on the position of a charge in an electric field the sign of work done by a force o a + charge is a hill when you go up the hill, the V and U are positive and the farther they travel the bigger the magnitude o a - charge is a valley when you go down the valley, the V and U are - and the farther they travel the bigger the magnitude o V - potential gets smaller when it goes with the E field o U - potential energy gets smaller when it gets to go where it wants
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work done is conservative only matters on a to b, not on how it got there work done is + if it goes with the E field
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workd done = (charge)(E field)(distance traveled) when the force is conservative
U is potential energy change in energy is - when the work done is positive and something falls to a smaller potential
starting kinetic + potential = ending kinetic + potential electric potential energy between two point charges o particles, whether + or -, want to accelerate toward lower potential energy
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where r is the distance between them if the charges have the same sign, then the potential is +, if not,
this equation is a shared property of both point charges (i.e. there is no electric potential energy of A point charge without a field) works for spherically symmetric charge distributions (because of Gaussian surfaces, can treat a as its center because that is where all the charge seems to be concentrated anyways) electric potential energy between more then two point charges
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for total potential between a few point charges o
this is not a vector sum, it is just numbers ex
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electric POTENTIAL (not energy, just potential) o potential energy that would be associated with a unit charge at a certain point (potential energy per unit charge)
electric potential (Volt) = (potential energy)/(charge) = J/C
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work done per unit charge by an electric force from a to b = (potential at a) (potential at b) = potential of a with respect to b this is VOLTAGE! the difference between the potential at a and the potential at b
electric potential relates to E
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o o
we do initial - final here, where a is initial and b is final
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electric field units
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equipotential surfaces o like contour lines
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because potential energy does not change as a test charge moves over an equipoteitnal surface, the E field doesn't do any work when E is big, the equipotential surfaces are closer together because the E does a lot of work on a test charge in that small distance o field lines and equipotential surfaces are ALWAYS perpendicular o conductors and equipotential surface at rest the surface is always an equipotential surface no potential difference between two points on the surface, V is constant the insides all have the same potential difference, 'equipotential volume' the surface charge density on the wall of the cavity is 0 at every point aka, a hollow metal sphere will hav no charge on the insides, just the outsides Potential Gradient o used when V is known, but we want to find E o
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points in the direction V increases, so E is in the direction V decreases always perpendicular to the equipotential surface (duh)
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24 Capacitance and Dielectrics •
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capacitor o any two (usually charged) conductors separated by a distance with an insulator (air and vacuums count) between o the conductors are called electrodes o close enough that the Q on one side = the -Q on the other net charge remains 0 this is the ideal situation, but in rl its not always so o stores energy o has to be induced (have work done on it), like a battery the voltage difference between a capacitor and a battery for one of each is the same for both the capacitor and the battery capacitance
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capacitance = charge on one of the conductors/the potential difference between them o SI unti - Farad = 1F = 1C/V o a measure of a capacitor's ability to store energy o for any capacitor in a vacuum, the capacitance only depends on the shapes, dimensions, and separation of the conductors parallel-plate capacitor
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capacitors in: o series
connected by a single, straight wire the charge magnitude on the plates are always the same (induced charges, all add up to 0), the potential difference of each capacitor is not necessarily the same. The individual potential differences add to give the total
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equivalent capacitance any number of capacitors lined up in series can be replaced by a single capacitor with the same capacitance
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the place matters, the charge is the same always less then individual capacitance of each capacitor
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the capacitors are equally far from the two points the potential difference for all individual capacitors is the same
equivalent capacitance any number of capacitors lined up in parallel can be replaced by a single capacitor with the same capacitance
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the charge matters, the place stays the same always greater then individual capacitance of each capacitor
general
charging
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when the switch is initially closed, current will rush to the capacitor, causing the bulb to glow as charge builds on the capacitor, the voltage starts opposing that of the battery: current drops therefore the bulb grows dimmer when the voltage of the capacitor ='s that of the battery, the plates are completely charged: current ='s 0, no brightness in bulb disconnected
absolutely nothing happens. the capacitor will ideally keep its charge forever because nothing can flow through it discharged
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all the charge goes back into the resistor or battery, loses all voltage causes a smaller current that can light the bulb momentarily energy storage of capacitors o the electric potential energy stored in a charged capacitor ='s the work required to charge it
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Q is in coulombs, C is in farads (C/V), V is in volts (J/C), U is in joules
Dielectrics o insulating sheets that separate the conductors in a capacitor o three reasons for it 1. solves the mechanical problem of having large metal sheets not touch 2. increases the max possible potential difference between the plates 3. by increasing the max possible potential difference, the actual potential difference decreases, increasing capacitance
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dielectric constant of a material
where C is with the dielectric, and C0 is the capacitance in a vacuum
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28 When the space between two plates is completely filled by the material, Q is constant
by looking at the E field according to the picture
dielectric breakdown it is possible for a material with a high dielectric constant (like water) to be a bad material. this is because slowly the electrons will dissolve in the material, causing the charge to flow between the capacitor and plates, just like a conductor even with normal used material, a high enough E field can rip electrons free from the dielectric called a short circuit o on a molecular level o
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Gauss's law in dielectrics
Qencl-free does not include induced charge
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27 Magnetic Field and Magnetic Forces •
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differences between Electric and Magnetic field o Electric field is created by charged objects and induction produces force if the object is charged o Magnetic field is created by moving electric charges (including currents in a wire) produces force if the object is charged AND the object had a velocity AND the velocity is in a different direction of the magnetic field B magnetic forces only act on moving charges magnetic poles act just like dipoles, except they cant appear seperatly o break a bar magnet apart, now there are 4 poles
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magnetic field o created by a current or moving point charge just as a resting point charges creates an electric field o creates a force F on any other moving charge or current just as a resting point charge creates a force F = qE magnetic force o
right hand rule v to B, F is thumb when q is -, F points in the opposite direction B magnetic field vector measured in Tesla: T: N/Am gauss: G: 10^-4 T showing F's direction on paper a dot . points out of the paper (arrow head) a x points into the paper (arrow feathers) o when both electric and magnetic field are present
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magnetic field lines behave just like electric field lines
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they DO NOT point in the direction of force as E lines do the force depends on the particles velocity and the charges sign o also, unlike electric field lines that end at a -, magnetic field lines never ever ever end, they form closed loops magnetic flux
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same as electric flux magnetic flux density flux per unit area across an area at right angles to the magnetic field
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motion of a charged particle in a magnetic field o if the magnetic field is the only force acting on the particle, the particle will always move with a constant speed
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adding an E field to a B field and applications o velocity selector o
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mass spectrometer
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magnetic force on a current carrying conductor o for a straight wire segment
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if it is curved, take the line integral:
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it doesn't matter if I is the flow of electrons or protrons
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Forces and Torques on a Current Loop
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force on the closed loop in a magnetic field F = IaB i force = (current)(length a)(magnetic field) in the x direction there is an opposing force -F, therefore the net force is 0 this is true for all current loops in a uniform magnetic field. NET FORCE = 0 torque on a closed loop in a magnetic field
torque = (current)(magnetic field)(area - ab)sin(angle between where the loop is and where the loop wants to be) the loop wants to go perpendicular to the magnetic field! if this loop is wirelicious (a solenoid aka coil) N is loops
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potential energy for a magnetic dipole (includes closed loops w/current in a magnetic field)
potential energy = -(current)(area)(magnetic field)cos(angle between)
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28 Sources of Magnetic Field •
Magnetic Field of a Moving Charge o
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magnetic field of current elements o
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similar to a moving particle because magnetic fields follow the law of superposition (integral in included, yay)
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for a straight current-carrying conductor
force between parallel conductors
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force per unit length = (constant)(current of 1)(current of 2)/(distance between) direction of force if the currents go in the same way, they snap together
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if the current goes opposite ways, they repel
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ampere
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the current that makes the force per unit length of two conductors at infinite L and 1 meter apart = 2* 10^-7 Magnetic field of a circular current loop o
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right hand rule gives direction magnetic field of a coil with N loops o
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Ampere's Law o
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works for any path you choose around a wire
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general ampre's law:
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current = emf/resistance an induced current occurs on a coil when there is a change of flux over a change of time. for example: o the B field changes o the moment when an electromagnet is turned on or off o the moment when a coil develops a velocity nonparallel to a steady B field a velocity in a changing B field will keep the flux changing, therefore the induced current is until that flux stops changing or the coil stops moving o the area of a coil grows or decreases by squeezing or stretching the 2pir^2 area o while the coil is being unwound or wound more o
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Faraday's law of induction
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flux reminder
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direction of induced emf o Faraday's Law: it is - for a + change in flux. this means that for the right hand rule, it goes the opposite way o Lenz's Law: the direction of any magnetic induction effect is such as to oppose the cause of the effect
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motional electromotive force o for a straight rod in a perpendicular B field a particle inside a moving rod in a magnetic field experience force from B the movement of FB creates an excess of charge on end end of the rod, which creates an electric force in equilibrium FE = FB
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this works for any closed conducing loop
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these equations are equivalent to Faraday's equation: episolon = (change in flux)/ (change in time) this form more useful when the conductor is moving
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emf (E) is measured in V Induced Electric Fields o occurs when a flux changes through a stationary conductor o ex:
current through wire changes, causing a change in flux which means there is an emf in the wire loop the question: what makes the charges move around in the wire loop? can't be magnetic force (the loop isn't in a a magnetic field ..?) must be an INDUCED ELECTRIC FIELD that causes it
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this induced E (nonelectrostatic) is not conservative around a closed loop like the regular E (electrostatic)
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direction o
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Displacement Current
obvious because
=/= 0
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with this fictitious 'displacement current', Ampere's law can be rewritten
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this is useful because it explains the 'current' that apparently runs through a capacitor. without it, Ampere's law contradicts itself
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Maxwell's equations
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the closed surface integral of Electric field ='s the enclosed charge/constant no charge enclosed = 0 surface integral of E
the closed surface integral of Magnetic field ='s 0 means that the magnetic field is conservative also shows that there are no magnetic monopoles (a magnetic field needs a north and south pole)
shows that both conduction current and displacement current act as a sources of magnetic field
shows that changing magnetic field of magnetic flux induces an electric field if the magnetic flux changes then the line integral is not 0 which means the electric field is not conservative this line integral must be carried out by a stationary path
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shows that tie-varying field of either kind induces a field of the other kind in neighboring regions of space such disturbances are called electromagnetic waves prove physical basis for light, radio, pretty much every wave
21 Electric Charge and Electric Field • •
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Electrostatics o interactions between electric charges that are at rest sign of charge related to attraction o two positively charge objects will repel same with negatively o a negative and a positive charge will always attract principle of electric charge o in a closed system, the sum of all electric chrages is constant an electron has the same charge magnitude as a proton o must be a whole number (can't have a fraction of a charge) transferring charges o conductor transfers charge has free electrons that are willing to travel o insulator does not transfer charge doesn't have many, or any, free electrons complicated molecules only a handful of elements (carbon, sulfur, phosphorus) are insulators rubbing 2 insulators together can result in one having a - charge while the other has a + o doesn't work for conductors induction o charging a body with the opposite sign of an object without losing the charge of the object o induced charge - areas of, say a ball, whose sides both have excess opposite charges o
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charged objects can attract an insulator
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Coulomb o the SI unit of electric charge o represents the magnitude of the charge of about 6*10^18 electrons an electron has a charge of 1.602*10^-19 Coulomb's law
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electric force between two point charges (N) = (constant)(|charge of one object * charge of the other object|)/(distance between them ^2) absolute values because the force can not be -, but the charges can constant usually = 9*10^9 (N m^2)/C^2 for 1 C charges. if 2 C, then multiply by 2 because epsilon-nought usually = 8.854*10^-12 C^2/(N m^2) o principle of superposition of forces Coulomb's law holds true for an infinite number of point charges ex. when two charges exert forces simultaneously on a third, the total force on that charge is the vector sum of the forces the two would exert individually force o equal and oppsoite (F[a on b] is - F[b on a])
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electric field o a body with a charge exerts an electric field whether another charge is there to feel it or not
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E is around A, test charge qo feels the force:
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F = ma = (charge)(electric field)
finding E
Electric Field Vector = (coonstant)(point charge)(unit vector between source point and field point; S and P)/(distance between the two)
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Vector field the varying from point to point of the electric field of a point charge for large objects, calculating the net electric force of point charges (infinite) on the body is the only way to accurately calculate the actual force
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Superposition of Electric Fields (non point charges)
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Electric Field Lines o demonstrates the direction of a system of electric fields o an individual electric field is tangent to any point on a line
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electric field lines bunched together means E is strong
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uniform electric field
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electric dipoles o pair of point charges with equal magnitudes and opposite signs (+/-) o example - water o
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the dipole makes it possible to lock molecules in place (Na+, Cl-) magnitude of dipole moment
magnitude of dipole moment (C*m) = (charge)(distance between them) the vector of p points toward the + o the net force on an electric dipole in a uniform external electric field always = 0 (not not the torque) o torque of a dipole while in an external electric field
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magnitude of torque = (magnitude of dipole moment)(magnitude of external electric field)(sin of the angle between p and E)
vector of torque on an electric dipole = cross product of p and E torque goes through the page (either out of in) and follows the right-hand rule in the direction of decreasing angle. the dipole (p) wants to be ||with E, AND pointing in the same direction in the example above, the torque is going into the page (rotating clockwise) o potential energy for a dipole in an electric field
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has a min when the angle = 0, pi, etc (||) has a max when the angle = pi/2, etc (+)
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22 Gauss's Law •
Flux o
the flow through an area
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where the box is imaginary
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calculating o V dot A where A is the area vector, V is the uniform velocity, or electric field, flowing through that area o
electric flux of a uniform, flat surface = electric field x area (cos(angle))
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flux is - if the A vector is choosen to point in the opposite direction of E 1D - linear charge denisty
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r^-2 2D - surface charge denisty
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= charge/area
r^-1 3D - volume charge denisty
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= charge/length
= charge/volume r^0 = 1
Gauss's Law o for a closed spherical surface o
electric flux = enclosed charge/constant independent of the radius o for any closed surface
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moral of the story: for any closed surface this in in theory. in practice, this is only true for symetircal surfaces o flux is - for a - inside charge (A is pointing the opposite way of E) o
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if the charge is outside the surface, the net flux is always 0 o
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where the chrages lie for solid conductors (metal ball) vs solid insulators (ball of yarn)
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conductor's inside - the net electric field inside a conductor is always zero. If the net electric field were not zero, a current would flow inside the conductor. This would build up charge on the exterior of the conductor. This charge would oppose the field, ultimately (in a few nanoseconds for a metal) canceling the field to zero.
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placing a charge inside a conductor o
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23 Electric Potential •
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electric potential o gravitational potential energy depends on the hight of a mass as electric potential depends on the position of a charge in an electric field the sign of work done by a force o a + charge is a hill when you go up the hill, the V and U are positive and the farther they travel the bigger the magnitude o a - charge is a valley when you go down the valley, the V and U are - and the farther they travel the bigger the magnitude o V - potential gets smaller when it goes with the E field o U - potential energy gets smaller when it gets to go where it wants
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work done is conservative only matters on a to b, not on how it got there work done is + if it goes with the E field
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reg
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workd done = (charge)(E field)(distance traveled) when the force is conservative
U is potential energy change in energy is - when the work done is positive and something falls to a smaller potential
starting kinetic + potential = ending kinetic + potential electric potential energy between two point charges o particles, whether + or -, want to accelerate toward lower potential energy
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where r is the distance between them if the charges have the same sign, then the potential is +, if not,
this equation is a shared property of both point charges (i.e. there is no electric potential energy of A point charge without a field) works for spherically symmetric charge distributions (because of Gaussian surfaces, can treat a as its center because that is where all the charge seems to be concentrated anyways) electric potential energy between more then two point charges
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for total potential between a few point charges o
this is not a vector sum, it is just numbers ex
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electric POTENTIAL (not energy, just potential) o potential energy that would be associated with a unit charge at a certain point (potential energy per unit charge)
electric potential (Volt) = (potential energy)/(charge) = J/C
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work done per unit charge by an electric force from a to b = (potential at a) (potential at b) = potential of a with respect to b this is VOLTAGE! the difference between the potential at a and the potential at b
electric potential relates to E
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o o
we do initial - final here, where a is initial and b is final
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electric field units
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equipotential surfaces o like contour lines
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because potential energy does not change as a test charge moves over an equipoteitnal surface, the E field doesn't do any work when E is big, the equipotential surfaces are closer together because the E does a lot of work on a test charge in that small distance o field lines and equipotential surfaces are ALWAYS perpendicular o conductors and equipotential surface at rest the surface is always an equipotential surface no potential difference between two points on the surface, V is constant the insides all have the same potential difference, 'equipotential volume' the surface charge density on the wall of the cavity is 0 at every point aka, a hollow metal sphere will hav no charge on the insides, just the outsides Potential Gradient o used when V is known, but we want to find E o
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points in the direction V increases, so E is in the direction V decreases always perpendicular to the equipotential surface (duh)
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24 Capacitance and Dielectrics •
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capacitor o any two (usually charged) conductors separated by a distance with an insulator (air and vacuums count) between o the conductors are called electrodes o close enough that the Q on one side = the -Q on the other net charge remains 0 this is the ideal situation, but in rl its not always so o stores energy o has to be induced (have work done on it), like a battery the voltage difference between a capacitor and a battery for one of each is the same for both the capacitor and the battery capacitance
o
capacitance = charge on one of the conductors/the potential difference between them o SI unti - Farad = 1F = 1C/V o a measure of a capacitor's ability to store energy o for any capacitor in a vacuum, the capacitance only depends on the shapes, dimensions, and separation of the conductors parallel-plate capacitor
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capacitors in: o series
connected by a single, straight wire the charge magnitude on the plates are always the same (induced charges, all add up to 0), the potential difference of each capacitor is not necessarily the same. The individual potential differences add to give the total
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equivalent capacitance any number of capacitors lined up in series can be replaced by a single capacitor with the same capacitance
o
the place matters, the charge is the same always less then individual capacitance of each capacitor
parallel
the capacitors are equally far from the two points the potential difference for all individual capacitors is the same
equivalent capacitance any number of capacitors lined up in parallel can be replaced by a single capacitor with the same capacitance
o
the charge matters, the place stays the same always greater then individual capacitance of each capacitor
general
charging
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when the switch is initially closed, current will rush to the capacitor, causing the bulb to glow as charge builds on the capacitor, the voltage starts opposing that of the battery: current drops therefore the bulb grows dimmer when the voltage of the capacitor ='s that of the battery, the plates are completely charged: current ='s 0, no brightness in bulb disconnected
absolutely nothing happens. the capacitor will ideally keep its charge forever because nothing can flow through it discharged
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all the charge goes back into the resistor or battery, loses all voltage causes a smaller current that can light the bulb momentarily energy storage of capacitors o the electric potential energy stored in a charged capacitor ='s the work required to charge it
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Q is in coulombs, C is in farads (C/V), V is in volts (J/C), U is in joules
Dielectrics o insulating sheets that separate the conductors in a capacitor o three reasons for it 1. solves the mechanical problem of having large metal sheets not touch 2. increases the max possible potential difference between the plates 3. by increasing the max possible potential difference, the actual potential difference decreases, increasing capacitance
o o
dielectric constant of a material
where C is with the dielectric, and C0 is the capacitance in a vacuum
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28 When the space between two plates is completely filled by the material, Q is constant
by looking at the E field according to the picture
dielectric breakdown it is possible for a material with a high dielectric constant (like water) to be a bad material. this is because slowly the electrons will dissolve in the material, causing the charge to flow between the capacitor and plates, just like a conductor even with normal used material, a high enough E field can rip electrons free from the dielectric called a short circuit o on a molecular level o
o
Gauss's law in dielectrics
Qencl-free does not include induced charge
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27 Magnetic Field and Magnetic Forces •
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differences between Electric and Magnetic field o Electric field is created by charged objects and induction produces force if the object is charged o Magnetic field is created by moving electric charges (including currents in a wire) produces force if the object is charged AND the object had a velocity AND the velocity is in a different direction of the magnetic field B magnetic forces only act on moving charges magnetic poles act just like dipoles, except they cant appear seperatly o break a bar magnet apart, now there are 4 poles
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magnetic field o created by a current or moving point charge just as a resting point charges creates an electric field o creates a force F on any other moving charge or current just as a resting point charge creates a force F = qE magnetic force o
right hand rule v to B, F is thumb when q is -, F points in the opposite direction B magnetic field vector measured in Tesla: T: N/Am gauss: G: 10^-4 T showing F's direction on paper a dot . points out of the paper (arrow head) a x points into the paper (arrow feathers) o when both electric and magnetic field are present
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magnetic field lines behave just like electric field lines
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o o
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they DO NOT point in the direction of force as E lines do the force depends on the particles velocity and the charges sign o also, unlike electric field lines that end at a -, magnetic field lines never ever ever end, they form closed loops magnetic flux
o o o
same as electric flux magnetic flux density flux per unit area across an area at right angles to the magnetic field
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motion of a charged particle in a magnetic field o if the magnetic field is the only force acting on the particle, the particle will always move with a constant speed
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adding an E field to a B field and applications o velocity selector o
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mass spectrometer
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magnetic force on a current carrying conductor o for a straight wire segment
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o
if it is curved, take the line integral:
o
it doesn't matter if I is the flow of electrons or protrons
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Forces and Torques on a Current Loop
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o
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force on the closed loop in a magnetic field F = IaB i force = (current)(length a)(magnetic field) in the x direction there is an opposing force -F, therefore the net force is 0 this is true for all current loops in a uniform magnetic field. NET FORCE = 0 torque on a closed loop in a magnetic field
torque = (current)(magnetic field)(area - ab)sin(angle between where the loop is and where the loop wants to be) the loop wants to go perpendicular to the magnetic field! if this loop is wirelicious (a solenoid aka coil) N is loops
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o
potential energy for a magnetic dipole (includes closed loops w/current in a magnetic field)
potential energy = -(current)(area)(magnetic field)cos(angle between)
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28 Sources of Magnetic Field •
Magnetic Field of a Moving Charge o
o
o
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magnetic field of current elements o
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similar to a moving particle because magnetic fields follow the law of superposition (integral in included, yay)
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for a straight current-carrying conductor
force between parallel conductors
o o
force per unit length = (constant)(current of 1)(current of 2)/(distance between) direction of force if the currents go in the same way, they snap together
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if the current goes opposite ways, they repel
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ampere
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the current that makes the force per unit length of two conductors at infinite L and 1 meter apart = 2* 10^-7 Magnetic field of a circular current loop o
o o
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right hand rule gives direction magnetic field of a coil with N loops o
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o
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Ampere's Law o
o
works for any path you choose around a wire
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o
general ampre's law:
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Physics 7D
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current = emf/resistance an induced current occurs on a coil when there is a change of flux over a change of time. for example: o the B field changes o the moment when an electromagnet is turned on or off o the moment when a coil develops a velocity nonparallel to a steady B field a velocity in a changing B field will keep the flux changing, therefore the induced current is until that flux stops changing or the coil stops moving o the area of a coil grows or decreases by squeezing or stretching the 2pir^2 area o while the coil is being unwound or wound more o
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o •
Faraday's law of induction
o •
flux reminder
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direction of induced emf o Faraday's Law: it is - for a + change in flux. this means that for the right hand rule, it goes the opposite way o Lenz's Law: the direction of any magnetic induction effect is such as to oppose the cause of the effect
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motional electromotive force o for a straight rod in a perpendicular B field a particle inside a moving rod in a magnetic field experience force from B the movement of FB creates an excess of charge on end end of the rod, which creates an electric force in equilibrium FE = FB
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o
this works for any closed conducing loop
o
these equations are equivalent to Faraday's equation: episolon = (change in flux)/ (change in time) this form more useful when the conductor is moving
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o
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emf (E) is measured in V Induced Electric Fields o occurs when a flux changes through a stationary conductor o ex:
current through wire changes, causing a change in flux which means there is an emf in the wire loop the question: what makes the charges move around in the wire loop? can't be magnetic force (the loop isn't in a a magnetic field ..?) must be an INDUCED ELECTRIC FIELD that causes it
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this induced E (nonelectrostatic) is not conservative around a closed loop like the regular E (electrostatic)
o
direction o
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Displacement Current
obvious because
=/= 0
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with this fictitious 'displacement current', Ampere's law can be rewritten
o
this is useful because it explains the 'current' that apparently runs through a capacitor. without it, Ampere's law contradicts itself
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o •
Maxwell's equations
o
the closed surface integral of Electric field ='s the enclosed charge/constant no charge enclosed = 0 surface integral of E
the closed surface integral of Magnetic field ='s 0 means that the magnetic field is conservative also shows that there are no magnetic monopoles (a magnetic field needs a north and south pole)
shows that both conduction current and displacement current act as a sources of magnetic field
shows that changing magnetic field of magnetic flux induces an electric field if the magnetic flux changes then the line integral is not 0 which means the electric field is not conservative this line integral must be carried out by a stationary path
o
o
o
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56 symmetry
shows that tie-varying field of either kind induces a field of the other kind in neighboring regions of space such disturbances are called electromagnetic waves prove physical basis for light, radio, pretty much every wave
