Question

A battery with a \(12\,\text{V}\) emf has an initial charge of \(80\,\text{A h}\). If the potential across the terminals stays constant until battery is completely discharged, then this battery can deliver energy at the rate of \(120\,\text{W}\) for a time:

• A. \(16\,\text{h}\)
• B. \(8\,\text{h}\)
• C. \(4\,\text{h}\)
• D. \(5\,\text{h}\)

Hint:

Battery capacity in ampere-hour is given by: \[ \text{Capacity}=I\times t. \] First calculate current using \[ P=VI, \] then determine the operating time.

Answer 1

The Correct Option is B

Step 1: Calculate the total energy stored in the battery.
A battery with $12\,\text{V}$ emf and capacity $80\,\text{A\,h}$ stores:
\[ E_\text{total} = V \times Q_\text{charge} = 12\,\text{V} \times 80\,\text{A\,h} = 960\,\text{W\,h} \] (Since $\text{V} \times \text{A} \times \text{h} = \text{W}\,\text{h}$)

Step 2: Convert to consistent units for time.
We want time in hours, so keep energy in $\text{W\,h}$ and power in $\text{W}$.

Step 3: Identify the power delivered.
The battery delivers energy at a rate of $P = 120\,\text{W}$.

Step 4: Use the energy-power-time relationship.
\[ \text{Time} = \frac{\text{Energy}}{\text{Power}} = \frac{960\,\text{W\,h}}{120\,\text{W}} = 8\,\text{h} \]

Step 5: Verify via current approach.
$I = P/V = 120/12 = 10\,\text{A}$. Time $= 80\,\text{A\,h} / 10\,\text{A} = 8\,\text{h}$. Both methods agree.

Step 6: State the answer.
\[ \boxed{8\,\text{h}} \]