You place two charges a distance r apart. Then you double each charge and double the distance between the charges. How does the force between the two charges change?
a) The new force is twice as large.
b) The new force is half as large.
c) The new force is four times as large.
d) The new force is four times smaller.
e) The new force is the same.
I picked e). Is this the correct answer. If not, what is the right answer and why? Thank you in advance for your help.
e) The new force is the same.
The correct answer is b) The new force is half as large.
According to Coulomb's law, the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Let's denote the original charge as Q1, the doubled charge as Q2, the original distance as r1, and the doubled distance as r2.
The original force between the charges is given by:
F1 = (k * Q1 * Q2) / r1^2
where k is a constant.
After doubling each charge and the distance, the new force between the charges is given by:
F2 = (k * 2Q1 * 2Q2) / (2r1)^2
= (k * 4Q1 * Q2) / (4r1^2)
= (k * Q1 * Q2) / r1^2
= F1
As you can see, the new force F2 is equal to the original force F1. Therefore, the correct answer is e) The new force is the same.
Based on the given scenario, you have two charges placed at a distance "r" apart. When you double each charge and double the distance between them, you are effectively modifying the variables in the equation for the force between two charges, which is given by Coulomb's Law.
Coulomb's Law states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:
F = k * (q1 * q2) / r^2
where:
- F is the force between the charges,
- k is the electrostatic constant,
- q1 and q2 are the magnitudes of the charges, and
- r is the distance between the charges.
In this case, you double each charge and double the distance, which means the new values will be 2q and 2r, respectively.
Now, let's see what happens to the force:
F_new = k * ((2q) * (2q)) / (2r)^2
= k * (4q^2) / (4r^2)
= (k * q^2) / (r^2)
Comparing this with the initial force (F), we can see that the new force (F_new) is equal to the initial force (F). Therefore, the correct answer is e) The new force is the same.
The force between the two charges remains unchanged when you double each charge and double the distance between them since both the numerator and denominator increase by the same factor of 4 (2^2), canceling each other out.