a +35x10^-6 C point charge is placed 32 cm from an identical +32x10^-6 C charge. how much work would be required to move a +50.0x10^-6 C test charge from a point midway between them to a point 12 cm closer to either of the charges?
please show work.
q is each of our two charges
Q is our test charge
Left charge at x = 0
Right charge at x = .32 m
force due to left charge = k q Q/x^2
force due to right charge = -k q Q/(.32-x)^2
when x = .16, the middle, the sum of those two forces is zero. However as you move off center, the one nearer will push back harder. We move to x = .16+.12 = .28. At that point we are .32 -.28 = .04 from the right charge.
The integral of dr/r^2 = -1/r =-[1/Rend -1/Rbegin] = [1/Rbegin - 1/Rend]
Let's assume we move right (+x direction)
Work done against left charge (negative because it is pushing in the direction of motion so we are holding back) = - k q Q( 1/.16 -1/.28)
Work done against right charge = +k q Q(1/.04 -1/.16)
To calculate the work required to move the test charge, we can use the formula:
Work = force x distance
First, let's find the force between the two charges using Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
Where:
F is the force between the charges,
k is the electrostatic constant and its value is 9 * 10^9 Nm^2/C^2,
|q1| and |q2| are the magnitudes of the two charges, and
r is the distance between the charges.
Calculating the force:
|q1| = +35 * 10^-6 C
|q2| = +32 * 10^-6 C
r = 32 cm = 0.32 m
F = (9 * 10^9 Nm^2/C^2) * ((35 * 10^-6 C) * (32 * 10^-6 C)) / (0.32 m)^2
F = (9 * 10^9 Nm^2/C^2) * (1.12 * 10^-9 C^2) / (0.32 m)^2
F = (9 * 10^9 Nm^2/C^2) * 3.5 * 10^-18 C^2 / 0.1024 m^2
F = 3.164 * 10^-6 N
Now, let's find the distance the test charge needs to move:
Given: 12 cm = 0.12 m
Next, we can calculate the work needed to move the test charge from the midpoint to a point 12 cm closer to either of the charges:
Work = force x distance
Work = (3.164 * 10^-6 N) * (0.12 m)
Work = 3.7968 * 10^-7 J
Therefore, the work required to move the +50.0 × 10^-6 C test charge from a point midway between the charges to a point 12 cm closer to either charge is approximately 3.7968 × 10^-7 J.