Question

A charged capacitor and an inductor are connected at time $\mathit{t}\mathbf{=}\mathbf{0}$. In terms of the period T of the resulting oscillations, what is the first later time at which the following reach a maximum: (a)${\mathit{U}}_{\mathbf{B}}$, (b) the magnetic flux through the inductor, (c) $\mathit{d}\mathit{i}\mathbf{/}\mathit{d}\mathit{t}$
, and (d) the EMF of the inductor?

Short answer

1. The value of $U_B$is maximum at $T/4$.
2. The magnetic flux through the inductor is maximum at$T/4$.
3. The value of $di/dt$is maximum at$T/2$.
4. The EMF of the inductor is maximum at $T/2$.

Step-by-step solution

The given data

A charged capacitor and an inductor are connected at time $t=0$.

Understanding the concept of the LC circuit mechanism

In an LC circuit, the charging and discharging of the capacitor occurs periodically. The total energy of the circuit remains constant and is distributed as magnetic energy and electrical energy. The proportion of the two types of energies varies periodically.

a) Calculation of the time at which the magnetic field energy is maximum

The charged capacitor and the inductor are connected $t=0$. The charged capacitor starts discharging through the inductor. Thus, the magnetic energy of the inductor goes on increasing.

So, this energy is maximum when the capacitor is fully discharged. This happens at time$t=T/4$.

Hence, the value of time is $t=T/4$.

b) Calculation of the time at which the magnetic flux is maximum

The magnetic flux through the inductor is also maximum at$t=T/4$since the energy is maximum at this time.

Hence, the value of time is $t=T/4$.

c) Calculation of the time at which the rate of current is maximum

Then the current through the circuit starts flowing and goes on increasing as the capacitor starts charging again. The current rises till the capacitor is charged fully but with opposite polarity. This occurs at time$t=T/2$.

Hence $di/dt$, is maximum at time $t=T/2$.

d) Calculation of the time at which the EMF is maximum

The EMF of the circuit is also maximum at the time, when the current flow is maximum. Hence, the EMF is maximum at $T/2$.