Mention any two advantages of parallel combination of resistances. In the given circuit, find the value of current in each resistance and the total current in the circuit.
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In parallel circuits, the total current is the sum of the currents through each branch, and the voltage across all resistors is the same.
Advantages of Parallel Combination of Resistances:
1. Constant Voltage:
In a parallel combination, the voltage across each resistance is the same. This ensures that each device or component receives the same voltage, which is often required for uniform operation.
2. Effective Resistance Decreases:
In a parallel combination, the total or equivalent resistance decreases as more resistances are added. This allows for more efficient current flow through the circuit. The total resistance is always lower than the smallest resistance in the combination.
Given:
- Resistances: \( R_1 = 8 \, \Omega \), \( R_2 = 12 \, \Omega \), and \( R_3 = 24 \, \Omega \)
- Voltage: \( V = 8 \, \text{V} \)
- The resistors are in parallel, so the total resistance \( R_{\text{eq}} \) can be calculated using the formula:
\[
\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\]
Substituting the values:
\[
\frac{1}{R_{\text{eq}}} = \frac{1}{8} + \frac{1}{12} + \frac{1}{24}
\]
Finding the LCM of the denominators:
\[
\frac{1}{R_{\text{eq}}} = \frac{3}{24} + \frac{2}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4}
\]
Thus, the total resistance:
\[
R_{\text{eq}} = 4 \, \Omega
\]
Total Current in the Circuit:
Using Ohm’s law, the total current \( I \) is given by:
\[
I = \frac{V}{R_{\text{eq}}} = \frac{8}{4} = 2 \, \text{A}
\]
Current in Each Resistance:
Since the resistances are in parallel, the current in each resistor can be calculated using Ohm’s law:
\[
I_1 = \frac{V}{R_1} = \frac{8}{8} = 1 \, \text{A}
\]
\[
I_2 = \frac{V}{R_2} = \frac{8}{12} = \frac{2}{3} \, \text{A}
\]
\[
I_3 = \frac{V}{R_3} = \frac{8}{24} = \frac{1}{3} \, \text{A}
\]
Conclusion:
The total current in the circuit is \( 2 \, \text{A} \). The current through each resistor is:
- \( I_1 = 1 \, \text{A} \) through the 8 \( \Omega \) resistor
- \( I_2 = \frac{2}{3} \, \text{A} \) through the 12 \( \Omega \) resistor
- \( I_3 = \frac{1}{3} \, \text{A} \) through the 24 \( \Omega \) resistor
