Electromagnetic Induction

#### 11.2.1 Describe the production of an induced e.m.f by relative motion between a conductor and a magnetic field (motionally induced e.m.f.).

When a conductor is moved through a magnetic field an electric current is induced in the conductor. Which makes sense, before we discussed how a moving electric charge feels a force due to a magnetic field, by moving the conductor we are moving the charges… Faraday found that the strength of the induced e.m.f. was proportional to:

1. The speed of the movement
2. The strength of the magnetic field
3. The number of turns on the coil
4. The area of the coil.

#### 11.2.2 Derive the formula for the e.m.f. induced in a straight conductor moving in a magnetic field

The force on the electrons in wire due to a magnetic field is:

(1)
\begin{align} F = qvB \sin \theta \end{align}

Thus we can say the potential difference (induced e.m.f. ) between the two ends of a conductor of length l is equal is defined as:

(2)
\begin{align} \epsilon = \frac{E_p}{q} = \frac{work}{q} = \frac{F \dot d}{q} \end{align}
(3)
\begin{align} \epsilon = \frac{qvB \sin \theta \dot l}{q} = Blv \sin \theta \end{align}

If the angle between the conductor and the magnetic field is 90° then the formula simplifies to:

(4)
\begin{align} \epsilon = Blv \end{align}

This last equation is in your IB formula handbook.

#### 11.2.6 Explain how a motionally induced e.m.f. can be equated to a rate of change of magnetic flux.

Magnetic flux is defined at the magnetic field strength times the area swept out by a conductor. Or in simpler terms it can be thought of as the number of magnetic field lines (which don’t really exist…) passing through a region. Mathematically we define magnetic flux as:

(5)
\begin{align} \Phi = AB \cos \theta \end{align}

Where A is the area swept out, B is the magnetic field strength and θ is the angle between the direction of motion and the magnetic field lines. The unit of magnetic flux is the Weber, Wb.

The area swept out by a moving straight conductor (a wire) is:

(6)
\begin{align} A = l \Delta x \end{align}

Where l is the length of the wire and $\Delta x$is the distance the wire moves, substituting into the magnetic flux equation:

(7)
\begin{align} \Phi = l \Delta x B \cos \theta \end{align}

Therefore the change in magnetic flux per time is equal to, assuming the angle is zero:

(8)
\begin{align} \frac{\Delta \Phi}{\Delta t} = \frac{l \Delta x B \cos \theta}{\Delta t} = Blv \end{align}

Which we can recognize from above as the induced e.m.f.

(9)
\begin{align} \epsilon = -\frac{\Delta \Phi}{\Delta t} \end{align}

This last equation is called Faraday’s law.

The flux linkage is defined as the number of loops (N) multiplied by the induced e.m.f. If there is N number of wires passing through the magnetic field then we can write the induced e.m.f. as:

This last equation is in your IB formula handbook.

#### 11.2.7 State Lenz’s law.

The emf induced in an electric circuit always acts in such a direction that the current it drives around the circuit opposes the change in magnetic flux which produces the emf.

If you move a conductor through a magnetic field a small current will be generated. The current will create a magnetic field, the current will always be in the direction so as to generate a force in the opposite direction as the motion. Almost like inertia.