Question

A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key S1 (keeping S2 open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference Vc (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as q = Q[1 − e−t/RC] The charging current can be obtained by differentiating it and using `d/dx(e^(mx))` = memx. Consider the case when R = 20 kΩ, C = 500 μF and V = 10 V.

(i) The final charge on the capacitor, when key S₁ is closed and S₂ is open, is ______.

1. 5 μC
2. 5 mC
3. 25 mC
4. 0.1 C

(ii) For sufficient time the key S₁ is closed and S₂ is open. Now key S₂ is closed and S₁ is open. What is the final charge on the capacitor?

1. Zero
2. 5 mC
3. 2.5 mC
4. 5 μC

(iii) The dimensional formula for RC is ______.

1. [M L² T⁻³ A⁻²]
2. [M⁰ L⁰ T⁻¹ A⁰]
3. [M⁻¹ L⁻² T⁴ A²]
4. [M⁰ L⁰ T A⁰]

(iv) The key S₁ is closed and S₂ is open. The value of current in the resistor after 5 seconds, is ______.

1. `1/(2 sqrte)` mA
2. `sqrt e` mA
3. `1/sqrt e` mA
4. `1/(2e)` mA

OR

(iv) The key S₁ is closed and S₂ is open. The initial value of charging current in the resistor, is ______.

1. 5 mA
2. 0.5 mA
3. 2 mA
4. 1 mA

Answer 1

q = Q[1 − e^−t/RC]

R = 20 kΩ

C = 500 μF

V = 10 V

(i) The final charge on the capacitor, when key S₁ is closed and S₂ is open, is 5 mC.

Explanation:

The final charge on the capacitor, when the key S₁ is closed and S₂ is open:

Q = CV    ...(As the battery will charge the capacitor to its full capacity)

Q = 500 × 10⁻⁶ × 10

= 5 mC

(ii) Zero

Explanation:

The capacitor will fully discharge through the resistor since S₁ is open. Therefore, the capacitor zero final charge.

(iii) The dimensional formula for RC is [M⁰ L⁰ T A⁰].

Explanation:

[RC] = [M L² T⁻³ A⁻²][M⁻¹ L⁻² T⁴ A²]

= [M⁰ L⁰ T A⁰]

(iv) The key S₁ is closed and S₂ is open. The value of current in the resistor after 5 seconds, is `bbunderline(1/(2 sqrt e) mA)`.

Explanation:

q = Q[1 − e^−t/RC]

I = `(dq)/dt`

Time constant (RC) = 20 kΩ × 500 μF

= 10

I = `5 xx (e^(-5/10))/10`

= `1/(2sqrte)` mA

OR

(iv) The key S₁ is closed and S₂ is open. The initial value of charging current in the resistor, is 0.5 mA.

Explanation:

I = `(dq)/dt`

= `Q(0 - e^(-1//tau)(-1/tau))`

I = `q/tau e^(-t/tau)`

⇒ I = `(CV)/(RC) e^(-t/tau)`

= `V/R e^(-t/tau)`

= `I_0e^(-t/tau)`

I₀ = `V/R`

= `(10 V)/(20 kOmega)`

= 0.5 mA