What is the new resistance of a given wire has resistance r if its radius is halved and length is unchanged?
2 Answers
To find the new resistance when the radius of the wire is halved and the length remains the same, you need to consider the formula for the resistance of a wire:
\[ R = \rho \frac{L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.
The cross-sectional area \( A \) of a wire with radius \( r \) is \( \pi r^2 \). If the radius is halved, the new radius is \( \frac{r}{2} \), and the new cross-sectional area \( A' \) is:
\[ A' = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{A}{4} \]
So, the new resistance \( R' \) is:
\[ R' = \rho \frac{L}{A'} = \rho \frac{L}{\frac{A}{4}} = 4 \cdot \rho \frac{L}{A} = 4R \]
Thus, the new resistance is four times the original resistance \( r \).
\[ R = \rho \frac{L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.
The cross-sectional area \( A \) of a wire with radius \( r \) is \( \pi r^2 \). If the radius is halved, the new radius is \( \frac{r}{2} \), and the new cross-sectional area \( A' \) is:
\[ A' = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{A}{4} \]
So, the new resistance \( R' \) is:
\[ R' = \rho \frac{L}{A'} = \rho \frac{L}{\frac{A}{4}} = 4 \cdot \rho \frac{L}{A} = 4R \]
Thus, the new resistance is four times the original resistance \( r \).
DC polarization is a phenomenon that occurs in electrochemical systems, where the application of a direct current (DC) voltage to an electrode causes a change in the electrode's behavior and the electrochemical reactions occurring at that electrode. Here's a more detailed explanation:
### Definition
**DC Polarization** refers to the alteration of the electrochemical potential of an electrode due to the application of a DC voltage. This effect can impact various aspects of the electrode's performance, such as its corrosion rate, efficiency, and the overall electrochemical reaction.
### Key Concepts
1. **Electrode Potential**: When a DC voltage is applied to an electrode, it shifts the electrode's potential away from its equilibrium potential. This shift can either make the electrode more positive (anodic polarization) or more negative (cathodic polarization).
2. **Polarization Curve**: This is a graphical representation of the relationship between the electrode potential and the current density. It helps in understanding how the electrode responds to different levels of polarization.
3. **Ohmic Polarization**: This type of polarization is due to the resistance of the electrolyte or the conductive material between the electrode and the reference point. It causes a drop in the potential along the path of the current flow.
4. **Activation Polarization**: This occurs due to the energy barrier for the electrochemical reaction at the electrode surface. It often involves overcoming the activation energy required for the reaction to proceed.
5. **Concentration Polarization**: This happens when the concentration of reactive species at the electrode surface changes due to the electrochemical reaction. It is influenced by the rate of mass transport of the species to and from the electrode surface.
### Applications and Implications
- **Corrosion Studies**: In corrosion science, DC polarization is used to study how materials corrode under various conditions. The polarization curve can provide information about the corrosion rate and mechanisms.
- **Electroplating**: In electroplating processes, DC polarization helps in controlling the deposition rate and quality of the plated material.
- **Battery Technology**: In batteries, DC polarization can affect the efficiency and lifespan of the battery. Understanding polarization helps in designing better batteries with improved performance.
- **Electrochemical Sensors**: Polarization is used to optimize the performance of electrochemical sensors by adjusting the electrode potential to enhance sensitivity and selectivity.
### Practical Considerations
- **Measurement**: To measure DC polarization, a potentiostat is commonly used. It applies a controlled DC voltage and measures the resulting current to analyze the polarization effects.
- **Compensating for Polarization**: In practical applications, techniques such as using reference electrodes and adjusting operational parameters are employed to manage and mitigate the effects of DC polarization.
In summary, DC polarization is a critical concept in electrochemistry that describes how the application of a DC voltage affects the behavior of an electrode. Understanding and controlling DC polarization is essential for optimizing electrochemical processes and devices.
### Definition
**DC Polarization** refers to the alteration of the electrochemical potential of an electrode due to the application of a DC voltage. This effect can impact various aspects of the electrode's performance, such as its corrosion rate, efficiency, and the overall electrochemical reaction.
### Key Concepts
1. **Electrode Potential**: When a DC voltage is applied to an electrode, it shifts the electrode's potential away from its equilibrium potential. This shift can either make the electrode more positive (anodic polarization) or more negative (cathodic polarization).
2. **Polarization Curve**: This is a graphical representation of the relationship between the electrode potential and the current density. It helps in understanding how the electrode responds to different levels of polarization.
3. **Ohmic Polarization**: This type of polarization is due to the resistance of the electrolyte or the conductive material between the electrode and the reference point. It causes a drop in the potential along the path of the current flow.
4. **Activation Polarization**: This occurs due to the energy barrier for the electrochemical reaction at the electrode surface. It often involves overcoming the activation energy required for the reaction to proceed.
5. **Concentration Polarization**: This happens when the concentration of reactive species at the electrode surface changes due to the electrochemical reaction. It is influenced by the rate of mass transport of the species to and from the electrode surface.
### Applications and Implications
- **Corrosion Studies**: In corrosion science, DC polarization is used to study how materials corrode under various conditions. The polarization curve can provide information about the corrosion rate and mechanisms.
- **Electroplating**: In electroplating processes, DC polarization helps in controlling the deposition rate and quality of the plated material.
- **Battery Technology**: In batteries, DC polarization can affect the efficiency and lifespan of the battery. Understanding polarization helps in designing better batteries with improved performance.
- **Electrochemical Sensors**: Polarization is used to optimize the performance of electrochemical sensors by adjusting the electrode potential to enhance sensitivity and selectivity.
### Practical Considerations
- **Measurement**: To measure DC polarization, a potentiostat is commonly used. It applies a controlled DC voltage and measures the resulting current to analyze the polarization effects.
- **Compensating for Polarization**: In practical applications, techniques such as using reference electrodes and adjusting operational parameters are employed to manage and mitigate the effects of DC polarization.
In summary, DC polarization is a critical concept in electrochemistry that describes how the application of a DC voltage affects the behavior of an electrode. Understanding and controlling DC polarization is essential for optimizing electrochemical processes and devices.