Physics Electromagnetism - How to Solve: RC Circuit (Charging/Discharging, Time Constant) – IIT JEE Guide

How to Solve: RC Circuit (Charging/Discharging, Time Constant) – IIT JEE Guide

(For Students & Teachers – Ready-to-Record Script Included)

Introduction

"Master RC circuits, and you unlock 5–10 marks in IIT JEE—questions on charging/discharging, time constants, and graph analysis appear every year. Miss this, and you’re leaving easy marks on the table."

(On camera: Hold up a past JEE paper with an RC circuit question highlighted.) "This is a 4-mark question from JEE Main 2022. Today, you’ll learn how to solve it in under 2 minutes."

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand: 1. Ohm’s Law & Kirchhoff’s Voltage Law (KVL) – How voltage divides in a loop. 2. Exponential Functions – The shape of ( e^{-t/RC} ) and ( 1 - e^{-t/RC} ). 3. Basic Calculus (Differentiation/Integration) – Only for derivation (not required for solving problems in JEE).

(On camera: Flash a quick recap slide of Ohm’s Law and exponential graphs.)

KEY TERMS & FORMULAS

Key Terms

Formulas

(All formulas are MEMORISE THIS unless marked otherwise.)

1. Time Constant (τ)
   [
   \tau = R \times C
   ]
2. ( R ) = Resistance (Ω)
3. ( C ) = Capacitance (F)

4. Unit: Seconds (s)

5. Charging a Capacitor (Voltage vs. Time)
   [
   V_C(t) = V_0 (1 - e^{-t/\tau})
   ]

6. ( V_C(t) ) = Voltage across capacitor at time ( t )
7. ( V_0 ) = Supply voltage (battery voltage)
8. ( t ) = Time (s)

9. ( \tau = RC )

10. Discharging a Capacitor (Voltage vs. Time)
    [
    V_C(t) = V_0 e^{-t/\tau}
    ]

11. ( V_0 ) = Initial voltage across capacitor

12. Current During Charging/Discharging
    [
    I(t) = \frac{V_0}{R} e^{-t/\tau}
    ]

13. Current decreases exponentially in both cases.

14. Charge on Capacitor (Q = CV)

15. Charging: ( Q(t) = Q_0 (1 - e^{-t/\tau}) )
16. Discharging: ( Q(t) = Q_0 e^{-t/\tau} )

17. ( Q_0 = C V_0 )

18. Energy Stored in Capacitor
    [
    U = \frac{1}{2} C V^2
    ]
    (Given on exam sheet, but useful for energy-based questions.)

(On camera: Point to each formula and say:) "Memorise these 4 equations—they’re your weapons for every RC circuit problem."

STEP-BY-STEP METHOD

Step 1: Identify the Circuit Type

• Charging: Battery is connected, switch is closed.
• Discharging: Battery is disconnected, capacitor discharges through resistor.

Step 2: Find the Time Constant (τ)

[ \tau = R \times C ] - If multiple resistors/capacitors, find equivalent R/C first.

Step 3: Write the Correct Voltage/Current Equation

• Charging: ( V_C(t) = V_0 (1 - e^{-t/\tau}) )
• Discharging: ( V_C(t) = V_0 e^{-t/\tau} )

Step 4: Plug in Given Values

• Substitute ( t, R, C, V_0 ) into the equation.
• If asked for current, use ( I(t) = \frac{V_0}{R} e^{-t/\tau} ).

Step 5: Solve for the Unknown

• Isolate the variable (e.g., ( t, R, V )).
• Use logarithms if needed (e.g., ( e^{-t/\tau} = 0.5 ) → ( t = \tau \ln 2 )).

Step 6: Check Units & Reasonableness

• ( \tau ) should be in seconds.
• Voltage should never exceed ( V_0 ) during charging.
• Current should decrease over time.

(On camera: Demonstrate each step on a whiteboard with a simple circuit.)

WORKED EXAMPLES

Example 1 – Basic (Charging)

Problem: A 1000 Ω resistor and 100 μF capacitor are connected to a 10 V battery. Find the voltage across the capacitor after 0.1 s.

Solution: 1. Identify: Charging circuit. 2. Time Constant:
[
\tau = R \times C = 1000 \times 100 \times 10^{-6} = 0.1 \text{ s}
] 3. Voltage Equation:
[
V_C(t) = V_0 (1 - e^{-t/\tau}) = 10 (1 - e^{-0.1/0.1}) = 10 (1 - e^{-1})
] 4. Calculate:
[
e^{-1} \approx 0.3679 \implies V_C = 10 (1 - 0.3679) = 6.32 \text{ V}
]

What we did and why: - Used the charging formula because the battery is connected. - Calculated ( \tau ) first, then substituted into the exponential equation.

Example 2 – Medium (Discharging + Equivalent Resistance)

Problem: A 500 Ω resistor and 200 μF capacitor are initially charged to 12 V. They are then connected to a 1000 Ω resistor. Find the voltage after 0.2 s.

Solution: 1. Identify: Discharging through two resistors in series (500 Ω + 1000 Ω = 1500 Ω). 2. Time Constant:
[
\tau = R_{eq} \times C = 1500 \times 200 \times 10^{-6} = 0.3 \text{ s}
] 3. Voltage Equation:
[
V_C(t) = V_0 e^{-t/\tau} = 12 e^{-0.2/0.3} = 12 e^{-2/3}
] 4. Calculate:
[
e^{-2/3} \approx 0.5134 \implies V_C = 12 \times 0.5134 = 6.16 \text{ V}
]

What we did and why: - Recognized that discharging involves all resistors in the path. - Combined resistors first, then used the discharging formula.

Example 3 – Exam-Style (Disguised Question)

Problem (JEE Main 2020): A capacitor of capacitance ( C ) is charged to a potential ( V ). It is then connected to a resistor ( R ). The time taken for the charge to reduce to ( \frac{1}{e} ) of its initial value is: (A) ( RC ) (B) ( \frac{RC}{2} ) (C) ( 2RC ) (D) ( RC \ln 2 )

Solution: 1. Identify: Discharging problem. 2. Charge Equation:
[
Q(t) = Q_0 e^{-t/\tau}
] 3. Given: ( Q(t) = \frac{Q_0}{e} )
[
\frac{Q_0}{e} = Q_0 e^{-t/\tau} \implies e^{-1} = e^{-t/\tau}
] 4. Solve for ( t ):
[
-1 = -\frac{t}{\tau} \implies t = \tau = RC
] 5. Answer: (A) ( RC )

What we did and why: - Recognized that ( \frac{1}{e} ) implies ( t = \tau ). - Avoided overcomplicating—exponential decay simplifies neatly.

COMMON MISTAKES

(On camera: Hold up a red card for each mistake and say:) "This is a 2-mark error. Don’t let it happen to you."

EXAM TRAPS

(On camera: Show a past JEE question with a trap highlighted.) "Examiners love this trick—spot it, and you save 3 minutes."

1-MINUTE RECAP

(Speak naturally, as if to a friend the night before the exam.)

"Alright, listen up. RC circuits are all about two things: the time constant ( \tau = RC ), and exponential decay. Here’s the cheat sheet:

1. Charging? Use ( V = V_0 (1 - e^{-t/\tau}) ).
2. Discharging? Use ( V = V_0 e^{-t/\tau} ).
3. Current? Always ( I = \frac{V_0}{R} e^{-t/\tau} ).
4. Time to reach X% charge? Set ( V = X\% \times V_0 ) and solve for ( t ).
5. Multiple resistors? Combine them first—no shortcuts.

Memorise these 4 equations, and you’re golden. Now go crush that exam."

(On camera: Point to the formulas one last time and smile.) "You’ve got this."