7. Find the resistance of the resistor shown in the diagram given the resistivity of the material it is made from. a
7. Find the resistance of the resistor shown in the diagram given the resistivity of the material it is made from. a
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![## Problem 7: Resistance Calculation of a Tapered Resistor
### Question:
Find the resistance of the resistor shown in the diagram given the resistivity of the material it is made from.
### Diagram Explanation:
#### Diagram Description:
The diagram illustrates a resistor that is conical in shape, with a length \( L \). The points labeled \( a \) and \( b \) represent the ends of the resistor. The resistor tapers from a larger cross-sectional area at \( a \) to a smaller cross-sectional area at \( b \).
### Steps to Solve:
1. **Understand the Geometry**:
- The resistor has a tapered (conical) shape which affects its cross-sectional area along its length.
2. **Resistance Formula**:
- Resistance \( R \) of a material can be calculated using the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the resistor,
- \( A \) is the cross-sectional area.
3. **Variable Cross-Section**:
- Since the cross-sectional area varies from point \( a \) to point \( b \), it requires integration to find the effective resistance.
- Assume the cross-sectional area changes linearly from \( A_1 \) at \( a \) to \( A_2 \) at \( b \).
4. **Calculate Area at a Given Point**:
- At a distance \( x \) from the left end \( a \),
\[
A(x) = A_1 + \left(\frac{A_2 - A_1}{L}\right)x
\]
5. **Integrate to Find Resistance**:
\[
R = \int_0^L \frac{\rho}{A(x)} dx = \int_0^L \frac{\rho}{A_1 + \left(\frac{A_2 - A_1}{L}\right)x} dx
\]
6. **Solve the Integral**:
- This integral simplifies to:
\[
R = \frac{\rho L}{A_2 - A_1} \ln\left(\frac{A_2}{A_1}\right)
\]
### Conclusion:
Combine these concepts and steps to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79826512-1148-4736-b4a7-6e7f4ec13f28%2F693e1fe9-61a9-4d9f-8d1c-d7a6c09165e7%2Fm3ed19n_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem 7: Resistance Calculation of a Tapered Resistor
### Question:
Find the resistance of the resistor shown in the diagram given the resistivity of the material it is made from.
### Diagram Explanation:
#### Diagram Description:
The diagram illustrates a resistor that is conical in shape, with a length \( L \). The points labeled \( a \) and \( b \) represent the ends of the resistor. The resistor tapers from a larger cross-sectional area at \( a \) to a smaller cross-sectional area at \( b \).
### Steps to Solve:
1. **Understand the Geometry**:
- The resistor has a tapered (conical) shape which affects its cross-sectional area along its length.
2. **Resistance Formula**:
- Resistance \( R \) of a material can be calculated using the formula:
\[
R = \rho \frac{L}{A}
\]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the resistor,
- \( A \) is the cross-sectional area.
3. **Variable Cross-Section**:
- Since the cross-sectional area varies from point \( a \) to point \( b \), it requires integration to find the effective resistance.
- Assume the cross-sectional area changes linearly from \( A_1 \) at \( a \) to \( A_2 \) at \( b \).
4. **Calculate Area at a Given Point**:
- At a distance \( x \) from the left end \( a \),
\[
A(x) = A_1 + \left(\frac{A_2 - A_1}{L}\right)x
\]
5. **Integrate to Find Resistance**:
\[
R = \int_0^L \frac{\rho}{A(x)} dx = \int_0^L \frac{\rho}{A_1 + \left(\frac{A_2 - A_1}{L}\right)x} dx
\]
6. **Solve the Integral**:
- This integral simplifies to:
\[
R = \frac{\rho L}{A_2 - A_1} \ln\left(\frac{A_2}{A_1}\right)
\]
### Conclusion:
Combine these concepts and steps to
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