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Homework Help Forum: physics
Current Questions | Post a New Question | Answer this Question | Further Reading

Posted by leesa on Friday, February 22, 2008 at 3:32pm.

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 and explain. i still don't understand what Damon said. we didn't learn how to solve it with integrals. is there another way

physics - Damon, Friday, February 22, 2008 at 4:03pm
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 solve this problem, we can use Coulomb's Law to find the forces between the charges at different positions.

The force between two charges is given by:

F = k * q1 * q2 / r^2,

where F is the force between the charges, k is the electrostatic constant (8.99x10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.

First, let's find the force between the charges when the test charge is at the midpoint between them (x = 0.16 m):

F_left = k * q * Q / (0.16)^2,

F_right = -k * q * Q / (0.16)^2.

Since the charges have the same magnitude, q = q = 35x10^-6 C.

Now let's find the distance between the test charge and the right charge when it is moved closer by 12 cm:

Distance from right charge = 0.32 m - 0.28 m = 0.04 m.

Next, let's find the force between the charges when the test charge is 12 cm closer to the right charge (x = 0.28 m):

F_left = k * q * Q / (0.16)^2,

F_right = -k * q * Q / (0.04)^2.

To find the work done to move the test charge from the midpoint to a point 12 cm closer to either charge, we need to integrate the forces over the distance. However, since you mentioned that you haven't learned about integrals, we can simplify the problem.

To find the work done by the left charge, we can use the equation:

Work_done_left = - F_left * distance,

Since the force and distance are in the same direction.

Similarly, to find the work done by the right charge, we can use the equation:

Work_done_right = - F_right * distance,

Again, since the force and distance are in the same direction.

Substituting the values, we have:

Work_done_left = - (k * q * Q / (0.16)^2) * distance,

Work_done_right = - (-k * q * Q / (0.04)^2) * distance.

Now we can calculate the work done by each charge.

Finally, we sum the work done by both charges:

Total_work = Work_done_left + Work_done_right.

Please note that this solution is a simplified approach that does not involve specific integrals.

To solve this problem, we can use the concept of electric potential energy and work. Electric potential energy is the potential energy associated with the interaction between charges. The work done on a test charge to move it from one point to another is equal to the change in electric potential energy.

First, let's find the electric potential energy at the initial point (midway between the two charges). The electric potential energy is given by the equation:

U = k(q1)(q2)/r

Where U is the electric potential energy, k is the electrostatic constant (approximately 9 x 10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.

Let's denote the left charge as q1 (+35x10^-6 C) and the right charge as q2 (+32x10^-6 C).

U_initial = k(q1)(q2)/r_initial

Now, we need to find the electric potential energy at the final point (12 cm closer to one of the charges). Let's assume we move closer to the right charge.

The distance from the right charge when moving 12 cm closer is now 0.12 m (from 0.16 m initially).

U_final = k(q1)(q2)/r_final

To calculate the work done on the test charge, we need to find the difference in electric potential energy between the initial and final points:

ΔU = U_final - U_initial

Next, we can calculate the work done:

W = ΔU

Finally, we can substitute the values into the equations and calculate the work.

Remember to convert the charges and distances to standard units (Coulombs and meters) before performing the calculations.