Consider the following four possibilities for two point charges and choose the one(s) that do not change the magnitude of the electrostatic force that each charge exerts on the other:

A. Double the magnitude of each charge and double the separation between them.
B. Double the magnitude of each charge and reduce the separation between them to half of its initial value.
C. Double the magnitude of only one charge and double the separation between the charges.
D. Double the magnitude of only one charge and increase the separation between the charges by a factor of square root(2).

I'm not really sure about this problem. I believe A is correct? I think D might also be correct, but I can't work out the math using theoretical #'s to make sure I'm correct. Someone please help explain this to me.

a,d

(a)k•q1•q2/r² = k•2q1•2q2/(2r)²
(d)k•q1•q2/r² = k•2q1•q2/(√2r)²

The answer is A and D, you are correct.

To determine which possibilities do not change the magnitude of the electrostatic force, we can use Coulomb's Law, which states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

Let's analyze each possibility:

A. Double the magnitude of each charge and double the separation between them.
In this case, if we double the magnitude of each charge, the product of their magnitudes will increase by a factor of 2 * 2 = 4. Additionally, if we double the separation between them, the square of the distance will also be multiplied by 2 * 2 = 4. Therefore, the force will be affected by a factor of (4) / (4) = 1. Therefore, this possibility does not change the magnitude of the electrostatic force.

B. Double the magnitude of each charge and reduce the separation between them to half of its initial value.
In this case, if we double the magnitude of each charge, the product of their magnitudes will increase by a factor of 2 * 2 = 4. However, if we reduce the separation between them to half of its initial value, the square of the distance between them will be divided by (1/2)^2 = 4. Therefore, the force will be affected by a factor of (4) / (4) = 1. Therefore, this possibility also does not change the magnitude of the electrostatic force.

C. Double the magnitude of only one charge and double the separation between the charges.
In this case, if we double the magnitude of one charge, the product of their magnitudes will be doubled. However, if we double the separation between the charges, the square of the distance will be multiplied by 2 * 2 = 4. Therefore, the force will be affected by a factor of (2) / (4) = 1/2. This means the magnitude of the electrostatic force will be halved. Therefore, this possibility does change the magnitude of the electrostatic force.

D. Double the magnitude of only one charge and increase the separation between the charges by a factor of the square root of 2 (√2).
In this case, if we double the magnitude of one charge, the product of their magnitudes will be doubled. Additionally, if we increase the separation between the charges by a factor of √2, the square of the distance between them will also be multiplied by (√2)^2 = 2. Therefore, the force will be affected by a factor of (2) / (2) = 1. Therefore, this possibility does not change the magnitude of the electrostatic force.

From the analysis above, the correct answer is A and D.

To determine which possibilities do not change the magnitude of the electrostatic force between two charges, we need to consider the equation for the electrostatic force:

F = k * (|q₁| * |q₂|) / r²,

where F is the magnitude of the electrostatic force, k is the electrostatic constant, |q₁| and |q₂| are the magnitudes of the charges, and r is the separation between the charges.

Let's go through each possibility one by one:

A. Double the magnitude of each charge and double the separation between them.
In this case, we have |q₁| → 2|q₁|, |q₂| → 2|q₂|, and r → 2r.

Plugging these values into the equation for the electrostatic force, we get:

F = k * (|q₁| * |q₂|) / r²
= k * (2|q₁| * 2|q₂|) / (2r)²
= k * (4|q₁| * |q₂|) / 4r²
= k * (|q₁| * |q₂|) / r²
= F.

Therefore, in case A, the magnitude of the electrostatic force remains unchanged.

B. Double the magnitude of each charge and reduce the separation between them to half of its initial value.
Here, |q₁| → 2|q₁|, |q₂| → 2|q₂|, and r → (1/2)r.

Plugging these values into the equation for the electrostatic force, we get:

F = k * (|q₁| * |q₂|) / r²
= k * (2|q₁| * 2|q₂|) / ((1/2)r)²
= k * (4|q₁| * |q₂|) / (1/4)r²
= 16 * (|q₁| * |q₂|) / r²
≠ F.

Therefore, in case B, the magnitude of the electrostatic force changes.

C. Double the magnitude of only one charge and double the separation between the charges.
In this case, a single charge is being doubled, but the other charge remains the same. Therefore, the magnitude of the electrostatic force will change.

D. Double the magnitude of only one charge and increase the separation between the charges by a factor of square root(2).
Similarly, in this case, a single charge is being doubled, and the separation is being increased. So, the magnitude of the electrostatic force will change.

In summary, the possibilities that do not change the magnitude of the electrostatic force are A. Double the magnitude of each charge and double the separation between them.