A point charge with charge Q1 is held stationary at the origin. A second point charge with charge moves from the point (x1, 0) to the point (x2,x2 ).
What is the question? What is the second point charge (Q2)?
To find the work done in moving the second point charge from (x1, 0) to (x2, x2), we need to determine the electric potential difference and use it in conjunction with the formula for work done in an electric field.
The electric potential difference (∆V) between two points in an electric field is given by the equation:
∆V = V2 - V1
where V2 is the potential at point (x2, x2) and V1 is the potential at point (x1, 0).
To calculate V1, we need to determine the electric potential due to the first point charge at point (x1, 0). The electric potential (V) due to a point charge at a distance r from it is given by Coulomb's law:
V = k * Q1 / r
where k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2).
For the first point charge at the origin, the distance (r1) from point (x1, 0) is given by:
r1 = √(x1^2 + 0^2) = x1
Substituting these values into the equation, we get:
V1 = k * Q1 / r1 = k * Q1 / x1
To calculate V2, we need to determine the electric potential due to both point charges at point (x2, x2). The distance (r2) from this point to the first point charge at the origin is given by:
r2 = √(x2^2 + x2^2) = √2 * x2
The potential due to the first point charge (Q1) is the same as before:
V1 = k * Q1 / x1
The potential due to the second point charge (Q2) is given by:
V2 = k * Q2 / r2
Substituting these values into the equation, we get:
V2 = k * Q2 / (√2 * x2)
Now that we have the potential difference (∆V), we can calculate the work done (W) using the formula:
W = Q2 * ∆V
Substituting the value of ∆V, we get:
W = Q2 * (V2 - V1)
= Q2 * (k * Q2 / (√2 * x2) - k * Q1 / x1)
Simplifying this expression gives us the final answer for the work done in moving the second point charge from (x1, 0) to (x2, x2).