Four wires made of same material have different lengths and radii, the wire having more resistance in the following case is

Options

A. `l = 100 cm, r = 1 mm `
B. `l = 50cm, r = 2mm `
C. `l = 100cm, r =(1)/(2)mm`
D. `l=50cm, r=(1)/(2)mm`

Answer

To determine which wire has the highest resistance among the four given cases, we can use the formula for resistance:

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

where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material (constant for all wires since they are made of the same material),
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

The cross-sectional area \( A \) for a wire with radius \( r \) is given by:

\[ A = \pi r^2 \]

Thus, we can rewrite the resistance formula as:

\[ R = \frac{\rho L}{\pi r^2} \]

From this, we can see that resistance \( R \) is directly proportional to the length \( L \) and inversely proportional to the square of the radius \( r \). Therefore, we can express this relationship as:

\[ R \propto \frac{L}{r^2} \]

Now, let's analyze each case provided:

### Case A:
- Length \( L_A = 100 \) cm
- Radius \( r_A = \frac{1}{10} \) cm

Calculating \( R_A \):

\[ R_A \propto \frac{100}{(\frac{1}{10})^2} = \frac{100}{\frac{1}{100}} = 10000 \]

### Case B:
- Length \( L_B = 50 \) cm
- Radius \( r_B = \frac{2}{10} \) cm

Calculating \( R_B \):

\[ R_B \propto \frac{50}{(\frac{2}{10})^2} = \frac{50}{(\frac{2}{10})^2} = \frac{50}{\frac{4}{100}} = 1250 \]

### Case C:
- Length \( L_C = 100 \) cm
- Radius \( r_C = \frac{1}{20} \) cm

Calculating \( R_C \):

\[ R_C \propto \frac{100}{(\frac{1}{20})^2} = \frac{100}{\frac{1}{400}} = 40000 \]

### Case D:
- Length \( L_D = 50 \) cm
- Radius \( r_D = \frac{1}{20} \) cm

Calculating \( R_D \):

\[ R_D \propto \frac{50}{(\frac{1}{20})^2} = \frac{50}{\frac{1}{400}} = 20000 \]

### Summary of Results:
- \( R_A \propto 10000 \)
- \( R_B \propto 1250 \)
- \( R_C \propto 40000 \)
- \( R_D \propto 20000 \)

From the calculations, we can see that:

\[ R_C > R_D > R_A > R_B \]

### Conclusion:
The wire with the highest resistance is **Case C**.

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