Two point charges 8q and -2q are located at x=0 and x=l find the location on the x axis where electric field due to these two point charges is zero
To find the location on the x-axis where the electric field due to the two point charges is zero, we need to consider the principle of superposition. According to the principle of superposition, we can calculate the combined electric field at any point on the x-axis by adding the individual electric fields due to each point charge.
Let's say we want to find the point where the electric field is zero on the x-axis, denoted as 'x = d'.
The electric field due to a point charge can be calculated using Coulomb's law:
E = k * q / r^2,
where E is the electric field, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the point charge to the location where we want to calculate the electric field.
For the point charge 8q (located at x = 0):
E1 = k * (8q) / r1^2,
where r1 is the distance from the point charge 8q to the point 'x = d'.
For the point charge -2q (located at x = l):
E2 = k * (-2q) / r2^2,
where r2 is the distance from the point charge -2q to the point 'x = d'.
Since we want the combined electric field to be zero:
E1 + E2 = 0,
k * (8q)/r1^2 + k * (-2q)/r2^2 = 0.
Now, let's simplify the equation. Since we have two equations with two unknowns (r1 and r2), we can solve for their values.
k * (8q)/r1^2 = -k * (-2q)/r2^2,
(8q)/r1^2 = (-2q)/r2^2,
(8q)/(r1^2) = -(2q)/(r2^2).
Now, we can cross-multiply:
(8q) * (r2^2) = -(2q) * (r1^2),
8 * r2^2 = -2 * r1^2.
Divide both sides by 2:
4 * r2^2 = -r1^2.
Since we're looking for the distance, we can square root the equation to eliminate the square:
2 * r2 = -r1.
Now, substitute l - d for r2 (as it is the distance between the point charge -2q and the desired point x = d):
2 * (l - d) = -r1.
Solving for r1:
r1 = -2 * (l - d).
Since the distance can't be negative, we can remove the negative sign:
r1 = 2 * (l - d).
So, the location on the x-axis where the electric field due to these two point charges is zero is at a distance 'd' from the point charge 8q, given by r1 = 2 * (l - d).