Lengths and cross-sectional areas of four pieces of nichrome wire are (L,A), (2L, A), (L, 2A) and (2L, 2A), respectively. If the same voltage difference V is applied across their lengths, which of the wires will get the hottest in steady state?

Options

A. Wire (L,A)
B. Wire (2L,A)
C. Wire (L,2A)
D. Wire (2L,2A)

Answer

To determine which nichrome wire will get the hottest when the same voltage difference \( V \) is applied across their lengths, we need to analyze the resistance of each wire and how it relates to power dissipation. The power dissipated in a resistor (which generates heat) is given by the formula:

\[ P = \frac{V^2}{R} \]

Where:
- \( P \) is the power (heat generated),
- \( V \) is the voltage across the wire,
- \( R \) is the resistance of the wire.

### Step 1: Calculate the Resistance of Each Wire

The resistance \( R \) of a wire can be calculated using the formula:

\[ R = \frac{\rho L}{A} \]

Where:
- \( \rho \) is the resistivity of the material (constant for nichrome),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

Now, we will calculate the resistance for each of the four wires:

1. **Wire 1**: Length \( L \), Area \( A \)
\[ R_1 = \frac{\rho L}{A} \]

2. **Wire 2**: Length \( 2L \), Area \( A \)
\[ R_2 = \frac{\rho (2L)}{A} = \frac{2\rho L}{A} \]

3. **Wire 3**: Length \( L \), Area \( 2A \)
\[ R_3 = \frac{\rho L}{2A} = \frac{\rho L}{2A} \]

4. **Wire 4**: Length \( 2L \), Area \( 2A \)
\[ R_4 = \frac{\rho (2L)}{2A} = \frac{2\rho L}{2A} = \frac{\rho L}{A} \]

### Step 2: Compare the Resistances

Now we can summarize the resistances:

- \( R_1 = \frac{\rho L}{A} \)
- \( R_2 = \frac{2\rho L}{A} \)
- \( R_3 = \frac{\rho L}{2A} \)
- \( R_4 = \frac{\rho L}{A} \)

### Step 3: Determine Which Wire Has the Lowest Resistance

From the calculated resistances:

- \( R_1 = \frac{\rho L}{A} \)
- \( R_2 = \frac{2\rho L}{A} \)
- \( R_3 = \frac{\rho L}{2A} \)
- \( R_4 = \frac{\rho L}{A} \)

The lowest resistance is \( R_3 = \frac{\rho L}{2A} \).

### Step 4: Determine Which Wire Gets the Hottest

Since power \( P \) is inversely proportional to resistance \( R \) (as \( P = \frac{V^2}{R} \)), the wire with the lowest resistance will generate the most heat.

Thus, the wire that will get the hottest is **Wire 3** with dimensions \( (L, 2A) \).

### Final Answer

**Wire 3 (Length \( L \), Cross-sectional Area \( 2A \)) will get the hottest in steady state.**

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