# UY1: Faraday’s Law Of Induction & Lenz’s Law

Faraday’s law of induction states that the induced e.m.f. in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.

$$\epsilon =- \frac{d\Phi_{B}}{dt}$$

#### Example: E.M.F. and current induced in a loop

The magnetic field between the poles of the electromagnet is uniform at any time, but its magnitude is increasing at the rate of 0.020 T/s. The area of the conducting loop in the field is $120 \, \text{cm}^{2}$, and the total circuit resistance, including the meter and the resistor, is $5.0 \, \Omega$. The induced e.m.f. and the induced current in the circuit is given by:

\begin{aligned} \epsilon &= \frac{dB}{dt}A \\ &= 0.24 \, \text{mV} \end{aligned}

\begin{aligned} I &= \frac{\epsilon}{R} \\ &= 0.048 \, \text{mA} \end{aligned}

If we have a coil with N identical turns, and if the flux varies at the same rate through each turn, the total rate of change through all the turns is N times a large as for a single turn. If $\Phi_{B}$ is the flux through each turn, the total e.m.f. in a coil with N turns is:

$$\epsilon =- N \frac{d\Phi_{B}}{dt}$$

#### Example: Magnitude & Direction Of An Induced E.M.F.

A coil of wire containing 500 circular loops with radius 4.00 cm is placed between the poles of a large electromagnet, where the magnetic field is uniform and at an angle of $60^{\circ}$ with the place of the coil. The field decreases at a rate of 0.200 T/s. The magnitude of the induced e.m.f.:

\begin{aligned} \epsilon &=-500 (-0.2)(0.04^{2} \pi) \cos{30^{\circ}} \\ &= 0.435 \, \text{V} \end{aligned}

### Lenz’s Law

We can find the direction of an induced e.m.f. or current by using:

$$\epsilon =- \frac{d \Phi_{B}}{dt}$$.

Lenz’s law states that the direction of any magnetic induction effect is such as to oppose the cause of the effect.

Lenz’s law is directy related to energy conservation.

#### Example 1

Recall that:

\begin{aligned} \Phi_{B} &= \vec{B}.\vec{A} \\ &= BA \cos{\phi} > 0 \end{aligned}

Since $|\vec{B}|$ is increasing with time,

$$\frac{d\Phi_{B}}{dt} > 0$$

Therefore,

$$\epsilon =-\frac{d \Phi_{B}}{dt} < 0$$

#### Example 2

\begin{aligned} \Phi_{B} &= \vec{B}.\vec{A} \\ &= BA \cos{\phi} < 0 \end{aligned}

Since $|\vec{B}|$ is increasing with time,

$$\frac{d\Phi_{B}}{dt} < 0$$

Therefore,

$$\epsilon =-\frac{d\Phi_{B}}{dt} > 0$$