Electromagnetism - RLC Circuit

A series RLC circuit is used in a radio to tune an FM station broadcasting at 103.7 MHz. The resistance in the circuit is 10 ohms and the inductance is 2.0 microhenries. What is the best estimate of the capacitance that should be used?

A. 200 pF
B. 50 pF
C. 1 pF
D. 0.2 pF
E. 0.02 pF

(GR9677 #26)

Solution:

X_L = X_C
ωL = 1 / ωC
C = 1 / ω²L 

Given:
f  = 103.7 × 10⁶ Hz
ω = 2πf  = 2π × 103.7 × 10⁶ ≈ 2π × 10² × 10⁶ = 2π × 10⁸
ω² = (2π × 10⁸)² ≈ 4π² × 10¹⁶ 
L  = 2.0 microhenries = 2 × 10⁻⁶ H

C = 1 / (4π² × 10¹⁶ × 2 × 10⁻⁶) 

= 1 / (8π² × 10¹⁰)

Take π² ≈ 10

C  ≈ 1/8 × 10⁻¹¹ 
= 1.25 × 10⁻¹² 
= 1.25 pF

Answer: C

Electromagnetism - RLC Circuit

In the RLC circuit shown, the applied voltage is ε(t) = ε_m cos ωt For a constant ε_m, at what angular frequency ω does the current have its maximum steady-state amplitude after the transients have died out?

A. 1/RC
B. 2L/R
C. 1/√(LC)
D. √[(1/LC) − (R/2L)²]
E.  √[(1/RC)² − (L/R)²]

(GR9277 #81)

Solution:

I_max when X_L = X_C

→ ωL = 1 / ωC

ω = 1/√(LC)

Answer: C

Electromagnetism - RLC Circuit

An AC circuit consists of the elements shown above, with R = 10,000 ohms, L = 25 millihenrie and C an adjustable capacitance. The AC voltage generator supplies a signal with amplitude of 40 volts and angular frequency of 1,000 radians per second. For what value of C is the amplitude of the current maximized?

A. 4 nF
B. 40 nF
C. 4 μF
D. 40 μF
E. 400 μF

(GR0177 #38)

Solution:

I_max when X_L = X_C

→ ωL = 1 / ωC

C = 1 / ω²L 
= 1 / [(10³)² × 25 × 10⁻³]
= 1 / (25 × 10³)
= 40 × 10⁻⁶ 
= 40 μF

Answer: D