Question E5A06:
What is the magnitude of the circulating current within the components of a parallel LC circuit at resonance?  

Choices:  
A: It is at a minimum  
B: It is at a maximum  
C: It equals 1 divided by the quantity 2 times pi, times the square root of (inductance L multiplied by capacitance C)  
D: It equals 2 times pi, times the square root of (inductance L multiplied by capacitance C)    

Correct Answer is  B: It is at a maximum  

At resonance in a parallel LC (tank) circuit, the inductor and capacitor carry large, equal, and opposite reactive currents that circulate between them while largely canceling at the input. This makes the circulating current maximum at resonance.

Why the correct answer is correct (B)
- At resonance, $X_L = X_C$, so the inductor current $I_L = V/X_L$ and capacitor current $I_C = V/X_C$ are equal in magnitude and 180° out of phase.
- These branch currents cancel at the input, so the source current is minimal, but the currents within L and C themselves are large, limited mainly by losses (the circuit’s Q).
- In a high-Q tank, the circulating current is approximately:
  $$I_{\text{circ}} \approx Q \cdot I_{\text{source}} \quad \text{(at resonance)}$$
  so it is at a maximum at resonance.

Why the other choices are incorrect
- A: “It is at a minimum” — Incorrect. The minimum is the source current. The internal L and C currents are maximized at resonance.
- C: “It equals $1/(2\pi\sqrt{LC})$” — That expression is the resonant frequency $f_0$, not a current.
- D: “It equals $2\pi\sqrt{LC}$” — Wrong quantity and wrong units; this is not a current and is not related to circulating current magnitude.

Useful formulas and rules of thumb
- Resonant frequency:
  $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
- Reactances:
  $$X_L = 2\pi f L, \quad X_C = \frac{1}{2\pi f C}$$
- At parallel resonance (high Q), input impedance is maximum; approximate equivalent parallel resistance:
  $$R_p \approx Q\,X_L \quad (\text{high }Q)$$
- Current magnification at resonance (rule of thumb):
  $$I_{\text{circ}} \approx \frac{V}{X_L} \approx \frac{V}{X_C}, \quad \frac{I_{\text{circ}}}{I_{\text{source}}} \approx Q$$

Key idea: In a parallel LC at resonance, source current is minimized, but internal circulating currents are maximized.