An elastic circular conducting loop expands at a constant rate over time such that its radius is given by \(r(t)=r_{0}+v t\), where \(r_{0}=0.100 \mathrm{~m}\) and \(v=0.0150 \mathrm{~m} / \mathrm{s} .\) The loop has a constant resistance of \(R=12.0 \Omega\) and is placed in a uniform magnetic field of magnitude \(B_{0}=0.750 \mathrm{~T}\), perpendicular to the plane of the loop, as shown in the figure. Calculate the direction and the magnitude of the induced current, \(i\), at \(t=5.00 \mathrm{~s}\).

First, we need to find the radius of the loop at t=5s. We can use the given expression for the radius with respect to time: \(r(t)=r_{0}+vt\) \(r(5)=0.100\,\text{m} + 0.0150\,\text{m/s}(5\,\text{s})\) \(r(5)=0.100\,\text{m} + 0.0750\,\text{m} = 0.175\,\text{m}\) Now, we can calculate the area of the loop at t=5s: \(A=\pi r^2\) \(A=\pi (0.175\,\text{m})^2\approx 0.0962\,\text{m}^2\)

To calculate the magnetic flux through the loop, we can use the following expression: \(\Phi = B_{0}\cdot A\) \(\Phi = 0.750\,\text{T} \cdot 0.0962\,\text{m}^2 \approx 0.0722\,\text{Wb}\)

Faraday's law states that the induced electromotive force (EMF) is equal to the negative of the time derivative of the magnetic flux: \(EMF = -\frac{d\Phi}{dt}\) To determine the rate of change of the magnetic flux, we will first need to express the magnetic flux as a function of time: \(\Phi(t) = B_{0} \cdot A(t)\) Now, differentiate the magnetic flux with respect to time: \(\frac{d\Phi(t)}{dt} = B_{0}\frac{dA(t)}{dt}\) To find the rate of change of the area A(t), we need to differentiate the area formula with respect to time: \(A(t) = \pi \cdot r^2(t)\) \(\frac{dA(t)}{dt} = 2\pi\cdot r(t) \cdot \frac{dr(t)}{dt} = 2\pi\cdot r(t)\cdot v\) Now, substituting the given values for r(t) and v: \(\frac{dA(t)}{dt}=2\pi(0.175\,\text{m})(0.0150\,\text{m/s})\approx0.0164\,\text{m}^2/\text{s}\) Now, we can calculate the rate of change of the magnetic flux at t=5s: \(\frac{d\Phi(t)}{dt} = B_{0}\frac{dA(t)}{dt}=0.750\,\text{T}(0.0164\,\text{m}^2/\text{s})\approx0.0123\,\text{Wb/s}\) Finally, we can calculate the induced EMF using Faraday's law: \(EMF = -\frac{d\Phi}{dt} = - 0.0123\,\text{Wb/s}\) The negative sign indicates that the direction of the induced current opposes the change in the magnetic field.

Now that we have the value of the induced EMF, we can calculate the induced current using Ohm's law: \(i = \frac{EMF}{R}\) \(i = \frac{-0.0123\,\text{Wb/s}}{12.0\,\Omega}\approx -0.00103\,\text{A}\)

According to Lenz's law, the induced current will flow in such a direction as to oppose the change in the magnetic field. In this case, since the magnetic field is increasing as the loop expands, the induced current will flow in a clockwise direction.

The magnitude of the induced current at t=5s is approximately 0.00103 A, and it flows in a clockwise direction within the loop.