Question

Which statement about the phase relation between the electric and magnetic fields in an LC circuit is correct? a) When one field is at its maximum, the other is also, and the same for the minimum values. b) When one field is at maximum strength, the other is at minimum (zero) strength. c) The phase relation, in general, depends on the values of \(L\) and \(C\).

Short answer

Answer: In an ideal LC circuit, when one field is at maximum strength, the other is at minimum (zero) strength. They are 90 degrees out of phase.

Step-by-step solution

Understand LC circuits

An LC circuit consists of an inductor (L) and a capacitor (C) connected in series or parallel. Such circuits are used to produce oscillations and are involved in various applications such as radios and oscillators. In an ideal LC circuit, we assume no energy loss, and the energy oscillates back and forth between the electric field in the capacitor and the magnetic field in the inductor.

Voltage and Current characteristics in an LC circuit

In an LC circuit, the voltage across the capacitor (U) and the current through the inductor (I) can be described by the following sinusoidal equations: \(U(t) = U_0 \cos (\omega t + \phi_U)\) \(I(t) = I_0 \cos (\omega t + \phi_I)\) where \(U_0\) and \(I_0\) are the maximum voltage and current amplitudes, \(\omega\) is the angular frequency, and \(\phi_U\) and \(\phi_I\) are the phase shifts of the voltage and current, respectively.

Analyze and compare electric and magnetic field phases

We can compare voltage (electric field) and current (magnetic field) functions to determine their phase relation. We know that: - When the voltage is at its maximum (positive or negative) across the capacitor, the electric field is also at its maximum. - When the current through the inductor is at its maximum (positive or negative), the magnetic field is also at its maximum. Now let's look at the given statements: a) When one field is at its maximum, the other is also, and the same for the minimum values. This statement suggests that \(U(t)\) and \(I(t)\) have the same phase, which means \(\phi_U = \phi_I\). b) When one field is at maximum strength, the other is at minimum (zero) strength. This statement suggests that \(U(t)\) and \(I(t)\) are 90 degrees out of phase, which means either \(\phi_U - \phi_I = \pi/2\) or \(\phi_U - \phi_I = -\pi/2\) c) The phase relation, in general, depends on the values of \(L\) and \(C\). This statement suggests that the phase relation could vary depending on the inductor and capacitor values in the circuit.

Identify the correct statement

Now recall the equations for voltage and current in an LC circuit. The voltage across the capacitor is at its maximum positive value when the capacitor is fully charged, and the current through the inductor is zero. Conversely, the current through the inductor is at its maximum (positive or negative) when the voltage across the capacitor is zero. This is because energy is being completely transferred from one component to the other. Therefore, the phase relation between the electric and magnetic fields in an LC circuit is 90 degrees out of phase. It is clear now that the correct statement is: b) When one field is at maximum strength, the other is at minimum (zero) strength.