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Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P?
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Consider a uniformly charged circular disk of radius R lying in the zy...
Understanding Electric Potential
The electric potential \( V \) at a point due to a continuous charge distribution can be calculated by integrating the contributions from each infinitesimal charge element over the entire distribution.
Setup for the Problem
- Consider a circular disk of radius \( R \) with a uniform surface charge density \( \sigma \) lying in the zy-plane.
- The point \( P \) is located at a distance \( z \) along the z-axis from the center of the disk.
Electric Potential Contribution
- The potential \( dV \) due to an infinitesimal charge element \( dq \) on the disk at point \( P \) is given by:
\[
dV = \frac{1}{4\pi \epsilon_0} \frac{dq}{r}
\]
where \( r \) is the distance from the charge element to point \( P \).
- The distance \( r \) can be expressed as:
\[
r = \sqrt{z^2 + r'^2}
\]
where \( r' \) is the radial distance from the center of the disk to the charge element.
Integration Over the Disk
- The total charge \( dq \) can be expressed as:
\[
dq = \sigma \, dA = \sigma \, r' \, dr' \, d\theta
\]
where \( dA \) is the area element in polar coordinates.
- The potential \( V \) at point \( P \) is then:
\[
V = \int_0^{2\pi} \int_0^{R} \frac{\sigma \, r' \, dr' \, d\theta}{4\pi \epsilon_0 \sqrt{z^2 + r'^2}}
\]
After performing the integration, the expression for the electric potential simplifies to:
Final Expression for Electric Potential
\[
V = \frac{\sigma}{2\epsilon_0} \left( \sqrt{R^2 + z^2} - z \right)
\]
This expression accurately represents the electric potential \( V \) at point \( P \) above the uniformly charged circular disk.
The electric potential \( V \) at a point due to a continuous charge distribution can be calculated by integrating the contributions from each infinitesimal charge element over the entire distribution.
Setup for the Problem
- Consider a circular disk of radius \( R \) with a uniform surface charge density \( \sigma \) lying in the zy-plane.
- The point \( P \) is located at a distance \( z \) along the z-axis from the center of the disk.
Electric Potential Contribution
- The potential \( dV \) due to an infinitesimal charge element \( dq \) on the disk at point \( P \) is given by:
\[
dV = \frac{1}{4\pi \epsilon_0} \frac{dq}{r}
\]
where \( r \) is the distance from the charge element to point \( P \).
- The distance \( r \) can be expressed as:
\[
r = \sqrt{z^2 + r'^2}
\]
where \( r' \) is the radial distance from the center of the disk to the charge element.
Integration Over the Disk
- The total charge \( dq \) can be expressed as:
\[
dq = \sigma \, dA = \sigma \, r' \, dr' \, d\theta
\]
where \( dA \) is the area element in polar coordinates.
- The potential \( V \) at point \( P \) is then:
\[
V = \int_0^{2\pi} \int_0^{R} \frac{\sigma \, r' \, dr' \, d\theta}{4\pi \epsilon_0 \sqrt{z^2 + r'^2}}
\]
After performing the integration, the expression for the electric potential simplifies to:
Final Expression for Electric Potential
\[
V = \frac{\sigma}{2\epsilon_0} \left( \sqrt{R^2 + z^2} - z \right)
\]
This expression accurately represents the electric potential \( V \) at point \( P \) above the uniformly charged circular disk.
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Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P?
Question Description
Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P?.
Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P?.
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Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P?, a detailed solution for Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? has been provided alongside types of Consider a uniformly charged circular disk of radius R lying in the zy-plane with its center at the origin. The surface charge density of the disk is o. You need to calculate the electric potential V at a point P located at a distance z above the center of the disk along the z-axis. Which of the following expressions correctly represents the electric potential V at point P? theory, EduRev gives you an
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