Eddy Currents

Induction and energy transfer (12.9) — where the energy goes

• Before eddy currents, the textbook settles the energy question with a loop. Consider a rectangular loop ABCD of width $L$ and total resistance $R$ being pulled out of a region of uniform field $B$ with constant velocity $\vec{v}$.
• Diagram in words: a dashed rectangle marks the boundary of the field region (crosses inside for $\vec{B}$ into the page); the conducting loop straddles that boundary, part in and part out; an external force $\vec{F}$ pulls it to the right; a clockwise induced current $i$ is marked, with forces $F_1$ on the left arm still in the field and $F_2$, $F_3$ on the top and bottom arms.
• To maintain a constant velocity, an external force $\vec{F}$ must be applied to overcome the magnetic force of equal magnitude acting in the opposite direction. The rate of doing work is

$$P = \dfrac{W}{t} = \dfrac{F\times d}{t} = F v = \vec{F}\cdot\vec{v}$$

• As the loop moves right, the area lying within the field decreases, so the flux decreases; with $x$ the length still inside the field,

$$\phi_B = B\cdot A = BLx$$

• By Faraday's law the magnitude of the induced emf is

$$\lvert e\rvert = \dfrac{d\phi_B}{dt} = \dfrac{d}{dt}(BLx) = BL\dfrac{dx}{dt} = BLv$$

• and the induced current, whose direction is given by the right-hand rule, is

$$i = \dfrac{\lvert e\rvert}{R} = \dfrac{BLv}{R}$$

• The three segments of the loop still in the field feel deflecting forces $\vec{F_1}$, $\vec{F_2}$, $\vec{F_3}$ from $\vec{F} = i\vec{L}\times\vec{B}$. By symmetry $\vec{F_2}$ and $\vec{F_3}$ are equal and opposite and cancel. The remaining force $\vec{F_1}$ is directed opposite to the applied force, so $\vec{F} = -\vec{F_1}$ and

$$\lvert F_1\rvert = iLB\sin 90^\circ = iLB = \lvert F\rvert$$

• Substituting for $i$,

$$\lvert F\rvert = \left(\dfrac{BLv}{R}\right)LB = \dfrac{B^2L^2v}{R}$$

• so the mechanical power is

$$P = Fv = \dfrac{B^2L^2v^2}{R}$$

• The rate of production of heat energy in the loop is

$$P = i^2R = \left(\dfrac{BLv}{R}\right)^2R = \dfrac{B^2L^2v^2}{R}$$

• The two are exactly equal. Thus the work done in pulling the loop through the magnetic field appears as heat energy in the loop.

Eddy currents

• Now suppose the conducting loop is replaced by a solid conducting plate. The relative motion between the conductor and the magnetic field still induces a current in the plate, and we still meet an opposing force, so work must still be done to move the conductor with uniform velocity $v$.
• But the conduction electrons no longer follow one path as they did in the loop; they swirl about within the plate as if caught in an eddy of water. Such a current is called an eddy current, and it can be represented on a diagram by a single closed swirl.
• Definition for the board: the circulating currents induced in a metal block, when it is placed or moved in a changing magnetic field, are called eddy currents.
• The induced current in the plate is responsible for transferring mechanical energy into heat energy.

The swinging-plate demonstration

• A conducting plate free to rotate about a pivot at its top is allowed to swing down like a pendulum through the gap of a magnet.
• Each time the plate enters and leaves the field, a portion of its mechanical energy is transformed to heat energy. After several such swings there is no mechanical energy left with the pendulum, and the heat produced is dissipated in the solid plate, making it warm.
• This is exactly the board's question about a copper disc pendulum swinging between the poles of a magnet: eddy currents induced in the disc oppose its motion (Lenz's law), so it is quickly damped and comes to rest, and its mechanical energy is converted into heat energy in the disc.
• Eddy currents can be reduced by introducing discontinuity in the structure of the conductor plate — cutting slots in it, or laminating it — so that the closed paths available to the eddy currents are broken.

Applications of eddy currents

• Dead beat galvanometer: when a galvanometer is used for measuring current, its coil is wound on a light aluminium frame to make it dead beat, that is, to bring the coil quickly to rest. The motion of the metal frame in the magnetic field gives rise to eddy currents in the frame; these oppose the motion and bring the coil to rest quickly.
• Induction motor (speedometer): eddy currents are used to indicate the speed of a vehicle. A strong magnet is kept rotating at a rate set by the speed of the vehicle, inside an aluminium drum pivoted by means of a spring. Eddy currents produced in the drum make it turn in the direction of the rotating magnet, and a pointer attached to the drum indicates the speed on a calibrated scale.
• Electric brake (induction brake): when a train is to be stopped, the power rotating the axle is switched off and at the same time a stationary magnetic field is applied to the rotating drum. Strong eddy currents are set up in the drum, and the torque they produce opposes the rotation of the drum and hence of the axle, bringing the train to rest quickly and smoothly.
• Induction furnace: the metal to be melted is placed in a huge crucible and a high-frequency alternating current is passed through a surrounding coil. The rapidly varying magnetic field produces very strong eddy currents, and the heat produced is enough to melt the entire block of metal in a short time. This method is generally used to make alloys of different metals in vacuum.
• A board question asking for applications wants any two of these, each with its one-line mechanism.

Standard results

• Loop pulled out of a field: $\lvert e\rvert = BLv$, $i = \dfrac{BLv}{R}$, $F = \dfrac{B^2L^2v}{R}$, $P = Fv = i^2R = \dfrac{B^2L^2v^2}{R}$.
• Eddy currents are circulating currents in a bulk conductor in a changing field; they always oppose the relative motion and convert mechanical energy into heat.
• They are reduced by slotting or laminating the conductor.