A coil of resistance `100 Omega` is connected across a battery of `emf 6.0 V`. Assume that the heat developed in the coil is used to raise its temperature. If the heat capacity of the coil is `4.0 J K^(-1)`, how long will it take to raise the temperature of the coil by `15^@ C`?

Explanation

To solve the problem step by step, we will follow the outlined process:

### Step 1: Calculate the current flowing through the coil
Using Ohm's Law, we can find the current (I) flowing through the coil. The formula is:
\[ I = \frac{E}{R} \]
where:
- \( E = 6.0 \, V \) (emf of the battery)
- \( R = 100 \, \Omega \) (resistance of the coil)

Substituting the values:
\[ I = \frac{6.0}{100} = 0.06 \, A \]

### Step 2: Calculate the power developed in the coil
The power (P) developed in the coil can be calculated using the formula:
\[ P = I^2 R \]
Substituting the values:
\[ P = (0.06)^2 \times 100 = 0.0036 \times 100 = 0.36 \, W \]

### Step 3: Calculate the heat required to raise the temperature
The heat (Q) required to raise the temperature of the coil can be calculated using the formula:
\[ Q = C \times \Delta T \]
where:
- \( C = 4.0 \, J/K \) (heat capacity of the coil)
- \( \Delta T = 15 \, K \) (temperature increase)

Substituting the values:
\[ Q = 4.0 \times 15 = 60 \, J \]

### Step 4: Relate the heat to the power and time
The energy supplied by the battery over time can be expressed as:
\[ Q = P \times t \]
Rearranging this gives:
\[ t = \frac{Q}{P} \]

### Step 5: Substitute the values to find time
Substituting the values we calculated:
\[ t = \frac{60}{0.36} \]

Calculating this gives:
\[ t = 166.67 \, seconds \]

### Step 6: Convert seconds to minutes (if needed)
To convert seconds into minutes, we divide by 60:
\[ t = \frac{166.67}{60} \approx 2.78 \, minutes \]

### Final Answer
It will take approximately **166.67 seconds** or **2.78 minutes** to raise the temperature of the coil by \( 15^\circ C \).

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