PHY294H - Lecture 24

Recommended reading: FGT Chap. 31
Recommended problems: FGT 31.6, 31.7, 31.12, 31.13.

Faraday's law and Lenz' law

Faraday's law states that a time varying magnetic flux through a loop induces a voltage drop around the loop. The loop can be a real wire, in which current flow is induced, or is can be a loop in space, in which case there is no current flow. Faraday's law is,

$${\cal {E}} = \oint \vec{E}\cdot d\vec{s} = -{d\phi_B \over dt}$$ (1)

where $\phi_B$ is the magnetic flux,

$$\phi_B = \oint \vec{B}\cdot d\vec{A}$$ (2)

The surface A can be chosen in many ways, provided it is the correctly connected to the loop in which we are interested.

Lenz's law is just to do with the sign of the induced voltage in equation (1). The minus sign on the right hand side of this equation indicates that the induced emf opposes a change in magnetic flux.

Note that in circuits we have used Kirchhoff's loop law where the sum of the voltage drops around a closed loop is equal to zero. This is what we mean when we say a that the voltage is conservative. Faraday's law says that the sum of the voltage drops around a closed loop is equal to the rate of change of flux through the loop. This means that we cannot use Kirchhoff's loop law in circuits where there is a changing applied flux. These circuits are non-conservative. It is possible to generalise Kirchhoff's law to treat this case, however that is beyond the scope of what we will cover here.

From the definition of magnetic flux it is evident that we can produce a time varying flux by either varying the magnetic field with time, and/or by varying the area through which the magnetic field passes. For example consider the case of a uniform magnetic field. The magnetic flux through a loop in this field is given by, $\phi_B = B A cos(\psi)$. We can make B time dependent, or we can change the area of the loop A or we can change the orientation of the loop with respect to the magnetic field. A time dependence in any one of these variables leads to a time varying magnetic flux. Our first examples will consider motional emf where the flux is changed by moving wires or loops through a constant magnetic field.

Motional emf

In many applications of Faraday's law, an emf is induced by moving a wire or a loop (or coil) through a magnetic field. This is called motional emf.

Example - A moving wire

Consider a uniform and constant magnetic field, B, directed along the z-axis. Now consider moving a wire, which is directed along the y-direction, of length L at constant speed v along the x-direction. An emf is developed between the ends of this wire. To find this emf, consider a rectangular loop which is composed of a side of length L lying on the y-axis (centered at the origin), the moving piece of wire, and the two sides which join them to form the rectangle.. These two joining sides have length b = vt, where we assume that the piece of wire starts at the origin at time t=0. The rate of change of the flux is given by,

$${\cal{E}} = -{d\phi_B\over dt} = -B L {db \over dt} = -B L v$$ (3)

This emf is induced around the loop, however the only conducting part of the loop is the piece of wire. The charges in the piece of wire move until the voltage drop between the ends of the wire just balance the induced emf, ie. V_wire = BLv. This can also be understood from the Lorentz force law. Consider the wire to be composed of negatively charge carriers. The motion of these carriers in the magnetic field leads to the Lorentz force law F_B = -e v B. The charges build up at the ends of the wire until the induced electric field produces a force on the conductors which just balances the magnetic force. We then have,

$$e E = e {V_{wire}\over L} = e v B$$ (4)

which gives V_wire = BLv as found from Faraday's law.

Example - Moving a wire loop out of a field

Now consider a square loop of wire which lies in the x-y plane, and where each side has length L. Consider that the half space x<0 constains a constant and uniform magnetic, B, directed along the positive z-axis. Now consider the situation in which the square loop is initially within the B field and it is drawn out of the B field at velocity v along the x-axis. If the loop has resistance Rfind the induced current.

The rate of change of flux is given by,

$${\cal{E}} = -{d\phi_B \over dt} = B L v$$ (5)

The induced emf is then BLv. The current in the loop is thus,

$$i = {BLv \over R}$$ (6)

The direction of the current is to oppose the changing flux. Since the flux is decreasing, the current flows counterclockwise in the x-y plane. This produces an induced flux in the z-direction which opposes the changing flux induced by the motion of the loop out of the uniform B field. Note that the induced current is small if the resistance of the loop is large, while the induced current is large if the resistance is small.

Force and energy in motional emf - Magnetic drag

In cases where an induced current flows, eg. a conducting loop, there is dissipation in the loop. This energy loss must be equal to the work done by an external force. If we assume that the loop is moving at constant velocity, the external force must be balanced by an internal force. What is the origin of this internal force? The answer is that the induced current, I, is in a magnetic field, so it experiences a force $I d\vec{l}\wedge \vec{B}$, where $\vec{B}$ is the applied magnetic field. This force must equal to the applied force if the loop is to maintain a constant velocity. The force due to the induced current resists the motion of a wire or loop through the magnetic field and so acts as a magnetic brake. This force is called a magnetic drag force and has uses ranging from magnetic brakes to damping of unwanted oscillations.

Example - The magnetic drag on the loop discussed above

The induced current is in a magnetic field, so it experiences a force. Notice that we cannot use the dipole formulae here as the magnetic field is non-uniform. The forces on the segment outside the field is zero. The forces on the two sides which are partly in and partly out the field balance. The only unbalanced force is the force on the wire segment that is still in the magnetic field. This segment experiences a force equal to,

$$F = i L B = {B^2L^2v \over R} = F_{external}$$ (7)

The external force must balance this force in order to pull the loop out of the magnetic field at constant velocity. The external force thus provides power given by,

$$P_{external} = \vec{F}_{external}\cdot \vec{v} = i L B = {B^2L^2v^2 \over R} = {{\cal{E}}^2\over R}$$ (8)

The mechanical power is dissipated as resistive losses due to current flow in the loop. The smaller the resistance the larger the resistive losses.