An inductor L and a resistor R are connected in series with a direct current source of emf E. The maximum rate at which energy is stored in the magnetic field is
Question
An inductor L and a resistor R are connected in series with a direct current source of emf E. The maximum rate at which energy is stored in the magnetic field is
Solution
The energy stored in the magnetic field of an inductor is given by the formula:
U = 1/2 * L * I^2
where: U is the energy, L is the inductance, and I is the current.
The rate at which energy is stored is the derivative of the energy with respect to time, which is:
dU/dt = L * I * dI/dt
The maximum rate at which energy is stored occurs when the current is at its maximum. In a series RL circuit connected to a DC source, the maximum current is E/R, where E is the emf of the source and R is the resistance.
So, the maximum rate at which energy is stored is:
dU/dt_max = L * E/R * d(E/R)/dt
Since E and R are constants, their derivative with respect to time is zero, so:
dU/dt_max = L * E/R * 0 = 0
Therefore, the maximum rate at which energy is stored in the magnetic field is zero. This is because once the circuit reaches steady state (which happens instantly in an ideal DC circuit), the current is constant, so there is no change in the magnetic field, and hence no change in the energy stored in it.
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