Faraday and Lenz

Question 1

Maxwell’s Third Equation (Faraday’s Law) - Integral Form

Equation

Answer 1

∮ |E . d|l = - d/dt ∫|B.^n dA

Question 2

Maxwell’s Third Equation (Faraday’s Law) - Integral Form

Description

Answer 2

-a changing magnetic flux through a surface induces an emf in any boundary path of that surface, and a changing magnetic field induces a circulating electric field

Question 3

Faraday’s Law

Right Hand Rule

Answer 3

• to the determine the positive normal direction, the right hand rule is used
• fingers point in direction you go around the loop

Question 4

Lenz’s Law

Answer 4

• arises from the conservation of energy
• currents induced by changing magnetic flux flow in a direction in the direction so as to oppose that change in flux
• changing magnetic flux induced an electric field whether or not there exists a conducting path in which current may flow, Lenz’s Law tells you the direction of the circulation of the electric field around a specified path even if no conduction current actually flows along that path

Question 5

Induced Electric Fields

Field Lines

Answer 5

• electric field in Faraday’s law is an induced electric field
• net electric field at any point in space is the sum of all electric fields present, induced plus electrostatic
• induced electric field does not have any positive charges to diverge from or any negative charges to diverge to, divergence=0
• -if the divergence is zero then the induced electric field must form loops

Question 6

Electrostatic Fields and Induced Electric Fields

Conservative or Non Conservative

Answer 6

• electrostatic fields are conservative which means that the work done by an electrostatic field around a closed path is zero
• the induced electric field is not conservative it is like a battery driving charge around a circuit
• the path can be a physical material or just empty space, the induced electric field exists in either case

Question 7

Electromotive Force

Answer 7

-the line integral of an induced electric field around a complete circuit (the circulation) is equal to the work done in moving a unit charge around that circuit
-by definition this is the electromotive force of the circuit
ε = ∮|E . d|l

Question 8

Circulation (Vector Field and Electric Field)

Definition

Answer 8

-the circulation of a vector field is the line integral of that vector field around a closed path
circulation of |A = ∮|A . d|l
-circulation of an induced electric field is the energy given to each coulomb of charge as it moves as it moves around the circuit

Question 9

Magnetic Flux

Open and Closed Surfaces

Answer 9

• the magnetic flux across a closed surface is zero (Gauss’s Law)
• but for an open surface it can be non-zero

Question 10

How can magnetic flux through a surface change?

Answer 10

1) flux density of the magnetic field changes
2) angle between the magnetic field and the surface changes
3) area of the surface changes

Question 11

Solenoid

Definition

Answer 11

a coil wound into a tightly packed helix

Question 12

Faraday’s and Lenz’s Law

Solenoid

Answer 12

-each loop of the solenoid will generate an emf given by Faraday’s Law
-so the total emf across the terminals of the solenoid will be N times as large as that for one loop, where N is the total number of loops
-we define the flux through the solenoid to be N times the flux on one of the loops
ϕ = N ∫ |B.^n dA
ε = - dϕ/dt

Question 13

Circulation of an Electric Field

Equation

Answer 13

circulation = ∮|E . d|l

Question 14

Irrotational

Answer 14

-a vector field whose curl is zero is irrotational

Question 15

Curl of a Vector Field

Definition

Answer 15

-found by considering the circulation per unit are over an infinitesimal path around the point of interest:
|∇x|A = lim 1/ΔS * ∮|A . d|l
-the overall direction of curl represents the axis about which rotation is greatest
-the direction of rotation is given by the right hand rule

Question 16

Curl of an Electric Field

Definition

Answer 16

-curl is the ratio of the circulation around a path C to the are enclosed by that path as the area it encloses tends to 0
|∇x|E = lim 1/ΔS * ∮|E . d|l

Question 17

Curl

Electric Field Derivation

Answer 17

(|∇x|E).^n = lim 1/ΔA * ∮|E . d|l
-consider the z component of the curl by taking a square area dx by dy in the x-y plane
(|∇x|E).^k = lim 1/ΔA * ∮|E . d|l
-E varies with position, when moving along side dx, E=Ex(y), when moving along dy, E=Ey(x+dx), when moving along -dx, E=Ex(y+dy), when moving along -dy, E=Ey(x) ->
=lim 1/dxdy (Ex(y)dx + Ey(x+dx)dy - Ex(y+dy)dx - Ey(x)dy)
-group Ex and Ey terms
=lim ([Ex(y)-Ex(y+dy)]/dy + [Ey(x+dx)-Ey(x)]/dx)
-these are the definitions of partial derivatives:
=[-∂Ex/∂y + ∂Ey/dx] = [∂Ey/∂x - ∂Ex/∂y]
-repeat for x and y components:
|∇x|E = (∂Ez/∂y-∂Ey/∂z)^i + (∂Ex/∂z-∂Ez/∂x)^j + (∂Ey/∂x-∂Ex/∂Ex/∂y)^k

Question 18

Faraday’s and Lenz’s Law - Differential Form

Derivation

Answer 18

∮ |E . d|l = - d/dt ∫|B.^n dA
-the definition of curl is:
(|∇x|E).^n = lim 1/ΔA * ∮|E . d|l
-combine the two expressions:
(|∇x|E).^n = lim 1/ΔA * (-d/dt ∮|B.^n dA)
-in the limit that area tends to zero, B will be constant, we dont allow ΔA to change as a function of time so ΔA s cancel:
(|∇x|E).^n = -d/dt (|B.^n )
-using the product rule:
(|∇x|E).^n = -d/dt(|B).^n -d/dt(^n) |B
-as we dont allow the position of ΔA to change as a function of time, d/dt(^n)=0 :
(|∇x|E).^n = -d/dt(|B).^n
(|∇x|E).^n = - d|B/dt . ^n
-the .^n is on both sides, this means that in all directions the compent of curl of E and -d|B/dt is the same so:
|∇x|E = - d|B/dt

Question 19

Faraday’s and Lenz’s Law - Differential Form

Equation

Answer 19

|∇x|E = - d|B/dt

Question 20

Faraday’s Law and Lenz’s Law - Differential Form

Description

Answer 20

• a magnetic field that changes with time generates a circulating electric field
• the left side of the equation, |∇x|E, quantifies the tendency of the electric field to circulate around a point
• the right side of the equation, - d|B/dt, quantifies the rate of change of magnetic field with respect to time

Question 21

Faraday’s Law Integral Form - Solvable Problems

Answer 21

1) Given information about the changing magnetic flux, find the induced emf
2) Given the induced emf on a specified path, determine the rate of change of the magnetic field magnitude or direction or the area bounded by the path
3) In situations of high symmetry it is also possible to find the induced electric field given the rate of change of the magnetic field

Question 22

What kind of electric field is |E in Faraday’s Law?

Answer 22

• an induced electric field, not an electrostatic one
• |E in Faraday’s Law represents the induced electric field at each point along the path C, a boundary of the surface through which the magnetic flux is changing over time
• the path maybe through empty space or a physical material

Question 23

Faraday’s Law Differential Form - Solvable Problems

Answer 23

1) Given the magnetic field as a function of time, find the curl of the induced electric field
2) Given an expression for the induced vector electric field, determine the rate of change of the magnetic field
3) Also useful for deriving the electromagnetic wave equation

Question 24

Curl of Induced and Electrostatic Fields

Answer 24

• the curl of an electrostatic field is always zero, electric field lines diverge from positive charges and converge on negative so never loop back on themselves
• if a charge was moved along a closed loop path in an electrostatic field it would do work on the charge when it was moving in the direction of the field but you would have to do work whilst moving against the field the net work done would be zero
• for induced electric fields, field lines form loops so integrating |E.d|L around any boundary path for the surface through which |B is changing produces a non-zero result
• the faster the change in |B, the greater the magnitude of the curl