---
title: "Laplace: current i(t) with two capacitor branches"
subject: "Electrical Circuit Theory"
chapter: 4
key: "semester_3/electrical_circuit_theory:laplace-current-50v-10ohm-two-rc"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/laplace-current-50v-10ohm-two-rc
site: BCT Engineering
---

# Laplace: current i(t) with two capacitor branches

Electrical Circuit Theory · Chapter 4: Transient analysis in RLC circuit by Laplace Transform

- Asked in: 2068 Chaitra, 6 marks
- Video: [Using Mesh Analysis in the S-Domain (Laplace)](https://www.youtube.com/watch?v=nO4rOteH60s) — Zahi Haddad
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/laplace-current-50v-10ohm-two-rc
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-4

## Question

Using laplace transformation technique, find the expression for current i(t) in the network shown below for t > 0 when the switch is closed at t = 0. Assume zero initial charge across the capacitors.
[Figure: 50 V source with a switch (t = 0) and a 10 Ω resistor carrying current i, feeding two parallel branches: a capacitor (value not labelled in the scan) in series with a 5 Ω resistor, and a 5 Ω resistor in series with a 0.5 F capacitor]

## Short answer

Combine parallel RC branches to get equivalent series R = 12.5 Ω and C = 1 F. Source current in s-domain is 4/(s+0.08). Inverse transform gives i(t) = 4 e^(−0.08t) A, t > 0.

## Answer

**Assumption:** the unlabelled capacitor is 0.5 F, the same as the other branch (the scan does not show it). Both capacitors are initially uncharged. Source = 50/s.

#### Branch impedances
- Branch 1: Z₁ = 5 + 1/(0.5 s) = 5 + 2/s
- Branch 2: Z₂ = 5 + 2/s (identical)

The two identical branches in parallel:

Z_p = (5 + 2/s)/2 = 2.5 + 1/s

#### Source current
```text
I(s) = (50/s) / (10 + 2.5 + 1/s)
     = 50 / (12.5 s + 1)
     = 4 / (s + 0.08)
```

**i(t) = 4 e^(−0.08t) A, t > 0**

This is a single exponential because the circuit reduces to R_eq = 12.5 Ω in series with C_eq = 1 F: τ = 12.5 s.

Check: i(0⁺) = 50/12.5 = 4 A (both capacitors act as shorts, so 10 + 5∥5 = 12.5 Ω) ✓, i(∞) = 0 ✓. At t = 0.1 s, i = 3.968 A, which matches the numerical inverse Laplace transform.

If the unlabelled capacitor were 1 F instead: I(s) = 50(10s + 3)/(125s² + 45s + 2), giving i(t) = 3.874 e^(−0.0519t) + 0.126 e^(−0.308t) A.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is I(s)?
   A. 50/(s + 12.5)
   B. 50/(12.5s + 1) = 4/(s + 0.08)
   C. 10/(s + 0.08)
   D. 4/(s + 12.5)
   Answer: B. I = (50/s)/(10 + 2.5 + 1/s) = 50/(12.5 s + 1). (Key point: Source current)

2. What does the circuit reduce to, and what is i(0+)?
   A. 10 ohm with 1 F; i(0+) = 5 A
   B. 12.5 ohm with 0.5 F; i(0+) = 5 A
   C. 12.5 ohm with 1 F; i(0+) = 4 A
   D. 12.5 ohm with 1 F; i(0+) = 50 A
   Answer: C. With capacitors as shorts, R = 10 + 5 parallel 5 = 12.5 ohm; i(0+) = 50/12.5 = 4 A, with tau = 12.5 s. (Key point: Equivalent circuit)

## Related in Electrical Circuit Theory

- [Laplace: current supplied by source, 10 V](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-source-current-10v-1ohm-1h) — Electrical Circuit Theory, chapter 4
- [Laplace: switch a to b, capacitor current and voltage](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-switch-a-to-b-two) — Electrical Circuit Theory, chapter 4
- [Initial values with switch open to closed (10 Ω, 15 Ω, 2 F)](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-conditions-switch-open-10-15) — Electrical Circuit Theory, chapter 2
- [Initial values with 20 Ω shorted by switch, 4 H, 2 F](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-values-switch-open-20-30) — Electrical Circuit Theory, chapter 2, asked 2×
- [Laplace: capacitor current and voltage, 10 V, 10 μF](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-capacitor-current-voltage-10v-5ohm) — Electrical Circuit Theory, chapter 4, asked 3×
- [Parallel RLC with exponential current source, Laplace](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aparallel-rlc-exponential-current-laplace) — Electrical Circuit Theory, chapter 4