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,y2).
How much work is done by the electrostatic force on the moving point charge?
Express your answer in joules. Use k for Coulomb's constant ().
Compute how the distance from the origin changes. That would be from
r1 = x1 to
r2 = sqrt(x2^2 + y2^2)
The work done moving the charge is
-k q1*q2/[(1/r2 - 1/r1]
You need to provide values for q2, q1, and the locations to compute the work in Joules.
Why are the r's reciprocal??
To find the work done by the electrostatic force on the moving point charge, we can use the formula for work:
Work = ∫ F · ds
where F is the force and ds is the displacement vector.
The electrostatic force between two point charges is given by Coulomb's law:
F = k * (q1 * q2) / r^2
where k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.
In this case, q1 is held stationary at the origin, and q2 moves from (x1,0) to (x2,y2). So the distance r between the charges is:
r = sqrt((x2 - x1)^2 + y2^2)
Let's assume q2 is positive for simplicity. Then the work done by the electrostatic force can be expressed as:
Work = ∫ (k * (q1 * q2) / r^2) · ds
The differential displacement vector ds is given by:
ds = dx · i + dy · j
where i and j are the unit vectors along the x and y axes.
Substituting ds into the expression for work, we get:
Work = ∫ (k * (q1 * q2) / r^2) · (dx · i + dy · j)
Work = ∫ (k * (q1 * q2) / r^2) · dx + ∫ (k * (q1 * q2) / r^2) · dy
To evaluate these integrals, we need to express dx and dy in terms of one variable. Since the x-coordinate varies from x1 to x2, we can rewrite dx as:
dx = (x2 - x1) · du
where du is a differential variable that varies from 0 to 1.
Similarly, since the y-coordinate varies from 0 to y2, we can rewrite dy as:
dy = y2 · dv
where dv is a differential variable that varies from 0 to 1.
Substituting these expressions into the work formula, we get:
Work = ∫ (k * (q1 * q2) / r^2) · (x2 - x1) · du + ∫ (k * (q1 * q2) / r^2) · y2 · dv
Integrating over the range 0 to 1 for both du and dv, we can simplify the expression:
Work = (k * (q1 * q2)) · (x2 - x1) · ∫(0 to 1) du + (k * (q1 * q2)) · y2 · ∫(0 to 1) dv
The integrals over du and dv evaluate to 1:
Work = (k * (q1 * q2)) · (x2 - x1) + (k * (q1 * q2)) · y2
This is the expression for the work done by the electrostatic force on the moving point charge.
To find the work done by the electrostatic force on the moving point charge, we need to calculate the electrostatic force and then use it to calculate the work.
The electrostatic force between two point charges is given by Coulomb's law:
F = k * (q1 * q2) / r^2
Where:
- F is the electrostatic force between the charges
- k is Coulomb's constant (8.99 * 10^9 N m^2/C^2)
- q1 and q2 are the charges of the two point charges
- r is the distance between the charges
In this case, q1 is held stationary at the origin, so its position is (0,0). The second point charge is moving from the point (x1,0) to the point (x2,y2).
The distance between the charges, r, can be calculated using the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - 0)^2)
To calculate the work done, we use the formula:
Work = F * d
Where:
- Work is the work done by the force
- F is the force
- d is the displacement of the moving charge
Since the moving charge is moving from (x1,0) to (x2,y2), the displacement can be calculated as:
d = sqrt((x2 - x1)^2 + (y2 - 0)^2)
Now, to find the work done, we substitute the values into the formula:
Work = F * d
= (k * (q1 * q2) / r^2) * d
Plugging in the values and expressing the answer in joules:
Work = (8.99 * 10^9 N m^2/C^2) * (q1 * q2) / r^2 * d
= (8.99 * 10^9 N m^2/C^2) * (q1 * q2) / ((x2 - x1)^2 + y2^2) * sqrt((x2 - x1)^2 + y2^2)