Physics

Problem

A square coil of a single turn is made from a thin copper wire. At an instant

t=0

, the side length of the square is

20\,\text{cm}

. The coil is situated in a uniform magnetic field whose direction is perpendicular to the plane of the coil. The numerical data are as follows: • Initial side length:

a=20\,\text{cm}

• Rate of increase of each side length:

da/dt = 1.0\,\text{mm s}^{-1}

(side length increases uniformly) • Magnetic field at

t=0

B_0 = 0.30\,\text{T}

• Rate of decrease of magnetic field magnitude:

dB/dt = -0.020\,\text{T s}^{-1}

(uniform in time) • The coil is tilted so that the angle between its normal and the magnetic field is a constant

60^{\circ}

. Neglect the resistance of the wire and answer the following. a. (i) Calculate the rate of change of magnetic flux through the coil that is produced solely by the change in the area. b. (ii) Calculate the rate of change of magnetic flux that is produced solely by the change in the magnetic field magnitude. c. (iii) Determine the numerical value (magnitude) of the net induced emf in the coil at

t=0

. State clearly whether the two contributions assist or oppose each other. d. (iv) Using Lenz’s law, state the sense (clockwise or counter-clockwise when viewed along the field direction) of the induced current at

t=0

. Justify your answer.

Intuitive explanation

This problem asks you to find the induced EMF in a coil whose both the area and the magnetic field are changing with time. The key is recognizing that Faraday's law $\mathcal{E} = -\frac{d\Phi_B}{dt}$ depends on the total rate of change of magnetic flux, which comes from two independent sources: the shrinking field strength and the expanding coil area. You'll need to use the product rule to differentiate $\Phi_B = B(t) \cdot A(t) \cdot \cos(60°)$ , treating each factor's time dependence separately.

Intuitive explanation

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