Question

In the given circuit, find the equivalent resistance between A and B and potential difference between the ends of the 5 \(\Omega\) resistance.

Hint:

When calculating the equivalent resistance of resistors in parallel, use the formula \(\frac{1}{R_{\text{total}}} = \sum \frac{1}{R_i}\). In series, simply add the resistances.

Answer 1

The circuit involves resistors in series and parallel. First, the two 4 \(\Omega\) resistors are in parallel. The total resistance of this parallel combination is: \[ \frac{1}{R_{\text{parallel}}} = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \] \[ R_{\text{parallel}} = 2 \, \Omega \] Now, this 2 \(\Omega\) combination is in series with the 5 \(\Omega\) resistor, so the total resistance of this combination is: \[ R_{\text{total1}} = 2 + 5 = 7 \, \Omega \] Next, this 7 \(\Omega\) is in parallel with the 10 \(\Omega\) resistor: \[ \frac{1}{R_{\text{total}}} = \frac{1}{7} + \frac{1}{10} \] \[ \frac{1}{R_{\text{total}}} = \frac{10 + 7}{70} = \frac{17}{70} \] \[ R_{\text{total}} = \frac{70}{17} \approx 4.12 \, \Omega \] So the equivalent resistance between A and B is approximately \(4.12 \, \Omega\). 
Now, using Ohm's law, the total current in the circuit is: \[ I = \frac{V}{R_{\text{total}}} = \frac{6}{4.12} \approx 1.46 \, \text{A} \] The potential difference across the 5 \(\Omega\) resistor can be found using: \[ V = I \times 5 = 1.46 \times 5 = 7.3 \, \text{V} \]