Induced emf in a Stationary Coil in a Changing Magnetic Field

The magnet-coil system

• A permanent bar magnet is mounted on an arc of a semicircle of radius 50 cm. The arc is part of a rigid aluminium frame and is suspended at the centre of the arc, so the whole system can oscillate freely in its own plane, like a pendulum.
• A coil of about 10000 turns of copper wire loops the arc so that the bar magnet can pass through the coil freely.
• When the magnet moves through the coil, the magnetic flux through the coil changes and an emf is induced. The coil itself never moves — this is the stationary-coil case.
• Diagram in words: a bar magnet fixed at the bottom of a light rigid arc pivoted at the centre of curvature above it; a many-turn coil straddling the arc at the lowest point (the mean position), so the magnet swings in and out through the coil.

Measuring the induced emf

• To measure the emf, a capacitor $C$ and a diode $D$ are connected across the coil. The induced emf charges the capacitor through the diode, and the voltage developed across the capacitor is measured.
• The capacitor may not charge up to the peak value in a single swing, because the time constant $RC$ can be larger than the time for which the emf is generated. It may take a few oscillations to charge the capacitor to the peak value, and a milliammeter (mA) in the circuit tells us when the charging current ceases to flow — that is, when the peak has been reached.

Shape of the $B$-$t$ and $e$-$t$ curves

• The magnet starts far from the coil, moves through it and recedes, so the flux through the coil starts small, rises to a maximum and falls again.
• There is a substantial magnetic field at the coil only when the coil is very near the magnet, and the magnet is fastest at the mean position of the oscillation, where the coil is. So $B$ changes slowly with time when the magnet is far away and rapidly when it approaches the coil.
• The graph of $B$ against $t$ is therefore a hump, labelled 'bcde'. The centre of the hump corresponds to the time when the magnet is inside the coil, and the flat portion 'cd' at the top corresponds to the finite length of the magnet.
• The flux is related to the field through a constant — the effective area, which is the number of turns multiplied by the area of the coil.
• The induced emf is proportional to $\dfrac{d\phi}{dt}$, that is to the slope of the $B$-$t$ curve. The slope is largest at the two times $t_1$ and $t_2$ on either side of the hump, so the emf is largest there.
• Lenz's law puts a minus sign in $e = -\dfrac{d\phi}{dt}$, so the emf is negative at $t_1$ (flux increasing) and positive at $t_2$ (flux decreasing). The $e$-$t$ graph is therefore one negative pulse followed by one positive pulse, both occurring during just half a cycle of the motion. On the return swing they are repeated, in the reverse order.

The charging circuit

• The diode conducts only during the positive pulse. On the first half swing the capacitor charges to some potential $e_1$; during the next half swing the diode is cut off until a positive pulse arrives, and then the capacitor charges to a slightly higher potential $e_2$.
• This continues for a few oscillations until the capacitor has charged up to the peak value $e_0$ of the emf pulse. At that stage the ammeter shows no further kick in the current.

Estimating the peak emf

• Write the induced emf using the chain rule, separating geometry from motion:

$$e = -\dfrac{d\phi}{dt} = -\dfrac{d\phi}{d\theta}\cdot\dfrac{d\theta}{dt}$$

• The first factor depends on the geometry of the magnet and the coil. At $\theta = 0$, the mean position, the flux $\phi$ is a maximum, so $\dfrac{d\phi}{d\theta}$ is actually zero there; we are interested in its maximum value, which occurs slightly off the mean position.
• The second factor comes from the oscillation equation. With $\theta_0$ the angular amplitude and $T$ the period ($\nu = \dfrac{1}{T}$),

$$\theta = \theta_0\sin 2\pi\nu t = \theta_0\sin\dfrac{2\pi t}{T}$$

$$\dfrac{d\theta}{dt} = \dfrac{2\pi\theta_0}{T}\cos\dfrac{2\pi t}{T}$$

• The peak emf $e_0$ corresponds to $\left(\dfrac{d\phi}{dt}\right)_{\max}$, which occurs at positions near the mean position; there the cosine does not differ much from unity. Hence

$$e_0 = \left(\dfrac{d\phi}{dt}\right)_{\max} = \left(\dfrac{d\phi}{d\theta}\right)_{\max}\cdot\dfrac{2\pi\theta_0}{T}$$

• Conclusion: for a given magnet-coil system the peak induced emf $e_0$ is directly proportional to the angular amplitude $\theta_0$ and inversely proportional to the time period $T$.

The simple stationary-coil calculation

• When the coil is fixed and the field is simply changed, the working is much shorter. For a coil of $N$ turns of constant area $A$ with $\vec{B}$ normal to its plane,

$$\lvert e\rvert = \left\lvert\dfrac{d(N\phi)}{dt}\right\rvert = N\dfrac{d(BA)}{dt} = NA\dfrac{dB}{dt} = NA\dfrac{B_{\text{final}}-B_{\text{initial}}}{\Delta t}$$

• Worked pattern from the textbook: 400 turns, each a square of side 20 cm, field changing linearly from 0 to 0.50 T in 0.8 s gives $\lvert e\rvert = 400\times(0.20)^2\times\dfrac{0.5}{0.8} = 10\ \text{V}$.
• If instead the flux is given as a function of time, differentiate it: for $\phi = 8t^2+6t+C$ mWb, $\lvert e\rvert = 16t+6$ mV, which is 38 mV at $t=2$ s.
• If the coil is closed, the induced current follows from $I = \dfrac{\lvert e\rvert}{R}$.

Standard results

• $\lvert e\rvert = NA\dfrac{dB}{dt}$ for a stationary coil of fixed area in a changing field.
• $e_0 = \left(\dfrac{d\phi}{d\theta}\right)_{\max}\dfrac{2\pi\theta_0}{T}$ for the oscillating magnet-coil system; $e_0\propto\theta_0$ and $e_0\propto\dfrac{1}{T}$.
• Two pulses, one negative then one positive, per half cycle of the magnet's motion.