Wire B has twice the length and twice the radius of wire A. Both wires are made from the same material. If wire A has a resistance R, what is the resistance of wire B?
Wire B has twice the length and twice the radius of wire A. Both wires are made from the same material. If wire A has a resistance R, what is the resistance of wire B?
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Question
![**Problem:**
Wire B has twice the length and twice the radius of wire A. Both wires are made from the same material. If wire A has a resistance R, what is the resistance of wire B?
**Explanation:**
The resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
For wire A:
- Resistance = \( R \)
- Length of wire A = \( L \)
- Radius of wire A = \( r \)
- Cross-sectional area of wire A = \( \pi r^2 \)
For wire B:
- Length of wire B = \( 2L \)
- Radius of wire B = \( 2r \)
- Cross-sectional area of wire B = \( \pi (2r)^2 = 4\pi r^2 \)
The resistance of wire B can be calculated as:
\[
R_B = \rho \frac{2L}{4\pi r^2} = \frac{\rho L}{2\pi r^2} = \frac{R}{2}
\]
Thus, wire B has a resistance of \( \frac{R}{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F440b7269-170b-4024-8d0e-4b8a2f594bf7%2Ff31120d5-b478-4a3f-b311-975f9136bc6b%2Fx8pts3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Wire B has twice the length and twice the radius of wire A. Both wires are made from the same material. If wire A has a resistance R, what is the resistance of wire B?
**Explanation:**
The resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
For wire A:
- Resistance = \( R \)
- Length of wire A = \( L \)
- Radius of wire A = \( r \)
- Cross-sectional area of wire A = \( \pi r^2 \)
For wire B:
- Length of wire B = \( 2L \)
- Radius of wire B = \( 2r \)
- Cross-sectional area of wire B = \( \pi (2r)^2 = 4\pi r^2 \)
The resistance of wire B can be calculated as:
\[
R_B = \rho \frac{2L}{4\pi r^2} = \frac{\rho L}{2\pi r^2} = \frac{R}{2}
\]
Thus, wire B has a resistance of \( \frac{R}{2} \).
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