Question

A rectangular conducting loop with dimensions \(a\) and \(b\) and resistance \(R\) is placed in the \(x y\) -plane. A magnetic field of magnitude \(B\) passes through the loop. The magnetic field is in the positive \(z\) -direction and varies in time according to \(B=B_{0}\left(1+c_{1} t^{3}\right),\) where \(c_{1}\) is a constant with units of \(1 / \mathrm{s}^{3}\). What is the direction of the current induced in the loop, and what is its value at \(t=1 \mathrm{~s}\) (in terms of \(a, b, R, B_{0},\) and \(\left.c_{1}\right) ?\)

Short answer

The direction of the induced current is clockwise around the loop (from above the positive z-axis), and its value at t=1s is given by the expression: $$ I(1) = \frac{3abB_{0}c_{1}^{2}}{R} $$

Step-by-step solution

Determine the direction of the induced current using Lenz's Law

According to Lenz's Law, the direction of the induced current will be such that it opposes the change in magnetic flux through the loop. Since the magnetic field is in the positive \(z\)-direction and increasing due to the given time dependence, this means the induced magnetic field aims to oppose the increase. The induced magnetic field will therefore be in the negative \(z\)-direction. To create this, the induced current will flow clockwise around the loop when viewed from above the positive \(z\)-axis.

Find the change in magnetic flux with respect to time (\(\frac{d \Phi}{dt}\))

The magnetic flux \(\Phi\) through the loop is given by the product of the area of the loop \(A\) and the magnetic field strength \(B\): $$ \Phi = A \cdot B $$ where \(A = ab\) is the area of the rectangle. The magnetic field \(B\) varies with time as \(B = B_{0}(1 + c_{1}t^{3})\). Differentiating both sides of the magnetic flux equation with respect to time, we get the rate of change of magnetic flux: $$ \frac{d\Phi}{dt} = \frac{d(ab \cdot B)}{dt} = ab \cdot \frac{dB}{dt} $$ Now we need to differentiate the time dependence of \(B\) given by \(B = B_{0}(1 + c_{1}t^{3})\). Using the chain rule, we get: $$ \frac{dB}{dt} = B_{0} \cdot c_{1} \cdot \frac{d}{dt}(1 + c_{1}t^{3}) = B_{0} \cdot 3c_{1}^{2}t^{2} $$ So, the rate of change of magnetic flux is: $$ \frac{d\Phi}{dt} = ab \cdot B_{0} \cdot 3c_{1}^{2}t^{2} $$

Calculate the induced EMF

According to Faraday's Law, the magnitude of the induced EMF is equal to the absolute value of the rate of change of magnetic flux: $$ |\varepsilon| = \left|\frac{d\Phi}{dt}\right| $$ For \(t = 1\text{ s}\), the induced EMF is: $$ |\varepsilon(1)| = |ab \cdot B_{0} \cdot 3c_{1}^{2}(1)^{2}| = 3abB_{0}c_{1}^{2} $$

Find the value of the induced current at \(t = 1 \text{ s}\)

Using Ohm's Law, the induced current is given by the ratio of induced EMF to the resistance of the loop \(R\): $$ I = \frac{\varepsilon}{R} $$ For \(t = 1\text{ s}\), the value of the induced current is: $$ I(1) = \frac{3abB_{0}c_{1}^{2}}{R} $$ So, the direction of the induced current is clockwise around the loop (from above the positive \(z\)-axis) and its value at \(t = 1\text{ s}\) is \(\frac{3abB_{0}c_{1}^{2}}{R}\).