Question

An ideal battery (with no internal resistance) supplies \(V_{\text {emf }}\) and is connected to a superconducting (no resistance!) coil of inductance \(L\) at time \(t=0 .\) Find the current in the coil as a function of time, \(i(t) .\) Assume that all connections also have zero resistance.

Short answer

Question: Determine the expression for the current in the superconducting coil as a function of time when connected to an ideal battery. Answer: \(i(t) = \frac{V_{emf}}{L}t\)

Step-by-step solution

Set up Kirchhoff's Voltage Law equation

For the given circuit, we can write Kirchhoff's Voltage Law (KVL) equation as: \(V_{emf} - L\frac{di(t)}{dt}=0\)

Solve the differential equation

To find \(i(t)\), we need to solve the first-order linear differential equation: \(L\frac{di(t)}{dt}=V_{emf}\) First, divide both sides by \(L\): \(\frac{di(t)}{dt}=\frac{V_{emf}}{L}\) Now, integrate both sides with respect to time to find \(i(t)\): \(\int \frac{di(t)}{dt} dt=\int\frac{V_{emf}}{L} dt\) On the left side, the integral of the derivative of \(i(t)\) is simply \(i(t)\), and on the right side, we integrate a constant with respect to time: \(i(t) = \frac{V_{emf}}{L}t+C\)

Determine the integration constant

At time \(t=0\), the current through the coil is zero, since the switch is closed at \(t=0\). Therefore, \(i(0) = 0\). Using this initial condition, we can find the integration constant \(C\): \(0 = \frac{V_{emf}}{L}(0) + C\) Solving for \(C\), we get \(C = 0\).

Write the final expression for \(i(t)\)

With the value of \(C\) determined, we can now write the final expression for \(i(t)\): \(i(t) = \frac{V_{emf}}{L}t\) This equation gives the current in the coil as a function of time when connected to an ideal battery with EMF \(V_{emf}\) and inductor with inductance \(L\).