Physics

Problem

A square coil of side

20\,\text{cm}

and

50

turns has a total resistance of

2.0\,\Omega

. It is placed with its plane perpendicular to a uniform magnetic field that varies with time as follows: • From

t=0

t=0.10\,\text{s}

, the field increases uniformly from

0.10\,\text{T}

0.60\,\text{T}

. • From

t=0.10\,\text{s}

t=0.30\,\text{s}

, the field decreases uniformly back to

0.20\,\text{T}

. a. (i) Sketch qualitatively the variation of magnetic field with time over the given interval, clearly marking numerical values on both axes. b. (ii) Determine the magnitude and direction of the induced emf during the first interval (

0\le t\le0.10\,\text{s}

). c. (iii) Obtain an expression for the induced emf during the second interval and calculate its value at

t=0.20\,\text{s}

. d. (iv) Compute the total heat produced in the coil during the entire

0.30\,\text{s}

period.

Intuitive explanation

The key is Faraday's law of electromagnetic induction: a changing magnetic flux through the coil induces an EMF, which drives a current. Since the field changes at a constant rate during each time interval, the rate of flux change $\frac{d\Phi}{dt}$ is constant in each segment, making the induced EMF constant and therefore the current constant too. Calculate the field's rate of change for each interval, find the induced EMF using $\mathcal{E} = N \frac{d\Phi}{dt}$ , then use Ohm's law to get the current.

Intuitive explanation

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