Two spheres of charge q1 and q2 that are equal in magnitude (i.e. |q1| = |q2|) are attached by a spring with a constant k = 1440kg/s2 and a rest length of x0 = 83.0cm. We take negative values of Äx for compression and positive Äx values for expansion. Knowing that q1 is positive and Äx = 4.40cm, what are the values of (a) q1 and (b) q2? If we know that q1 is negative and measure Äx = -3.50cm, then what are the values of (c) q1 and (d) q2? (hint for parts (c) and (d): take the magnitude of Äx under the square root sign and look at positive or negative roots for charges)

Question

Two spheres of charge q1 and q2 that are equal in magnitude (i.e. |q1| = |q2|) are attached by a spring with a constant k = 1440kg/s2 and a rest length of x0 = 83.0cm. We take negative values of Äx for compression and positive Äx values for expansion. Knowing that q1 is positive and Äx = 4.40cm, what are the values of (a) q1 and (b) q2? If we know that q1 is negative and measure Äx = -3.50cm, then what are the values of (c) q1 and (d) q2? (hint for parts (c) and (d): take the magnitude of Äx under the square root sign and look at positive or negative roots for charges)

Answer 1

To find the values of q1 and q2 in each given scenario, we can use the formula for the force between the two charged spheres given by Hooke's law:

F = k * (x - x0)

where F is the force, k is the spring constant, x is the displacement from the rest length, and x0 is the rest length.

(a) In the first scenario where q1 is positive and Δx = 4.40cm (compression):
The force on each sphere is the same, and we have F = q1 * q2 / (4πε₀r²), where ε₀ is the electric constant and r is the distance between the spheres.

Using Hooke's law, we can equate the force equation and solve for q1:
k * (Δx) = q1 * q2 / (4πε₀r²)

Given that q1 = q2, we can substitute q1 with q2 in the equation:
k * (Δx) = q2² / (4πε₀r²)

Rearranging to solve for q2:
q2² = k * (Δx) * (4πε₀r²)
q2 = √(k * (Δx) * (4πε₀r²))

Substituting the given values:
k = 1440 kg/s², Δx = 4.40 cm = 0.0440 m, r = x0 = 83.0 cm = 0.830 m, and ε₀ is the electric constant.

Now we can calculate q2:
q2 = √(1440 kg/s² * 0.0440 m * (4πε₀ * (0.830 m)²))

(b) To find q1 in the first scenario, we can substitute the value of q2 back into the equation:
k * (Δx) = q1 * q2 / (4πε₀r²)
q1 = (k * Δx * (4πε₀r²)) / q2

Now we can calculate q1 using the known values of k, Δx, r, and q2.

(c) In the second scenario where q1 is negative and Δx = -3.50cm (expansion):
Using the same approach as before, we can solve for q2 first:
q2 = √(k * (Δx) * (4πε₀r²))

Then, substituting the value of q2 back into the equation, we can find q1:
q1 = (k * Δx * (4πε₀r²)) / q2

(d) Now we can calculate q2 using the known values of k, Δx, r, and q1.

Remember to use the magnitude of Δx under the square root sign in both cases and consider the positive or negative roots for the charges depending on the given charge sign.