Electromagnetism - Capacitor

The capacitor shown in Figure 1 above is charged by connecting switch S to contact a. If switch S is thrown to contact b at time t = 0, which of the curve in Figure 2 above represents the magnitude of the current through the resistor R as a function of time?

A. A
B. B
C. C
D. D
E. E

(GR9677 #01)

Solution:

The capacitor is charged by connecting switch S to contact a, so the current after connecting switch to contact b (t = 0) must start with I = V/R.

Only plot A and B are right.

The current can not be the same all the time and since the voltage of a capacitor follows an exponential decay:

V(t) = V₀e^−t/RC 
 I(t) = V(t)/R = V₀e^−t/RC/ R

Only plot B is right.

Answer: B

Electromagnetism - High and Low-Pass Filters

The circuits below consist of two-element combinations of capacitors, diodes and resistors. V_in represents an ac-voltage with variable frequency. It is desired to build a circuit for which V_out ≈ V_in at high frequencies and V_out ≈ 0 at low frequencies. Which of the following circuits will perform this task?

(GR9677 #45)

Solution:

High-pass filter: V_out ≈ V_in at high frequencies, (ω → ∞)
Low-pass filter: V_out ≈ 0 at low frequencies (ω → 0)

Diode is a device to allow current flow in one direction only. A combination of Diode and Resistor will not affect the output frequency.
→ (B), (C) are FALSE

(A), (D), (E) circuits are combination of resistor and capacitor.

Capacitive reactance, X_C  = 1 / ωC
X_C → 0 for ω → ∞
X_C → ∞ for ω → 0

(A) FALSE
It is parallel combination of capacitor and resistor, voltage is uniform throughout the circuit.

(D) FALSE
It is Low-pass filter only.
Using Voltage Divider for series circuit: V_out = X_C / (R + X_C) V_in

X_C → 0 for ω → ∞

V_out = 0 / (R + 0) V_in ≈ 0

(E) TRUE
It is High and Low-pass filter.
Using Voltage Divider: V_out = R / (R + X_C) V_in

X_C → 0 for ω → ∞

V_out = R / (R + 0) V_in 
V_out ≈ V_in

X_C → ∞ for ω → 0
V_out = R / (R + ∞) V_in 
V_out ≈ 0

Answer: E

Notes: for similar problem see GR0177 #39

Electromagnetism - Oscilloscope

The circuit shown above is used to measure the size of the capacitance C. The y-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across R, and the x-coordinate of the spot is swept at a constant speed s. The switch is closed and then opened. One can then calculate C from the shape and the size of the curve on the screen plus a knowledge of which of the following?

A. V₀ and R
B. s and R
C. s and V₀
D. R and R'
E. The sensitivity of the oscilloscope

(GR9277 #86)

Solution:

The voltage of a capacitor follows an exponential decay:

V(t) = V₀e^−t/RC 

C depends on V(t), V₀, t, and R.

V₀ is given and known from the beginning of measurement.

The y-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across R. Thus, V(t) is the shape and the size of the curve on the screen.
 
The x-coordinate of the spot is swept at a constant speed s. From this we can calculate time t.

Therefore, we can calculate C from the shape and the size of the curve plus a knowledge of s and R.

Answer: B

Electromagnetism - High-Pass Filters

Which two of the following circuits are high-pass filters

A. I and II
B. I and III
C. I and IV
D. II and III
E. II and IV

(GR0177 #39)

Solution:

High-pass filter: V_out ≈ V_in  for frequency, ω → ∞

Inductive reactance, X_L = ωL
Capacitive reactance, X_C  = 1 / ωC

For ω → ∞
X_L → ∞
X_C → 0

Voltage divider:


Case I.

→ Low-pass filter


Case II.

Note: R ≪ ∞
→ High-pass filter


Case III.

→ High-pass filter


Case IV.

→ Low-pass filter

Answer: D