A linear charge of nonuniform density �E(x)=bx C/m, where b = 5.2 nC/m2, is distributed along the x-axis from 4.9 m to 6 m.
Determine the electric potential (relative to zero at infinity) of the point y = 8.6 m on the positive y-axis.
To determine the electric potential at a point, we can use the principle of superposition, which states that the total electric potential at a point due to multiple charges is the algebraic sum of the individual electric potentials caused by each charge.
In this case, we have a linear charge with nonuniform density, and we want to find the electric potential at a point on the positive y-axis. Since the linear charge is distributed along the x-axis, we need to find the electric potentials due to small segments of the linear charge and then integrate them to get the total electric potential at our desired point.
Let's break down the steps to find the electric potential at the point y = 8.6 m:
1. Define your variables:
- y: The distance along the y-axis (y = 8.6 m in this case)
- x: The position along the x-axis where the linear charge is located
2. Calculate the electric potential due to each small segment of the linear charge:
- The electric potential at a point P due to a linear charge with linear density λ(x) is given by:
V = k * ∫[from x1 to x2] (λ(x) / r) dx
where k is the electrostatic constant (k = 9.0 x 10^9 N m^2/C^2), r is the distance from the charge element to the point P, and λ(x) is the linear charge density function.
- In this case, the linear charge density function is given by λ(x) = bx, where b = 5.2 nC/m^2.
3. Determine the limits of integration:
- The linear charge is distributed along the x-axis from x1 = 4.9 m to x2 = 6 m.
4. Integrate the electric potential due to each charge segment:
- Substitute λ(x) = bx into the formula for electric potential, and evaluate the integral over the range [x1, x2].
5. Sum up the contributions from all charge segments:
- Since the electric potential follows the principle of superposition, sum up the electric potentials due to all charge segments.
6. Calculate the total electric potential at the desired point:
- Add up all the contributions from the charge segments to get the total electric potential at the point y = 8.6 m on the positive y-axis.
By following these steps, you can find the electric potential at the given point.