Two charges, qA and qB, are separated by a distance, d, and exert a force, F, on each other. Analyze Coulomb's law and answer the following questions.
(a) What new force will exist if qA is doubled?
(b) What new force will exist if qA and qB are cut in half?
(c) What new force will exist if d is tripled?
(d) What new force will exist if d is cut in half?
(e) What new force will exist if qA is tripled and d is doubled?
Coulomb's law states that the electric force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula for Coulomb's law can be expressed as:
F = k * (qA * qB) / d^2
Where:
F is the electric force between the charges,
k is the electrostatic constant (approximately 9 x 10^9 N*m^2/C^2),
qA and qB are the charges of the two particles, and
d is the distance between the charges.
Now, let's analyze each question:
(a) What new force will exist if qA is doubled?
To determine the new force if qA is doubled, we can plug the modified value of qA into the Coulomb's law formula. If qA is doubled, the new force can be calculated as:
F_new = k * (2qA * qB) / d^2
= 2 * (k * (qA * qB) / d^2)
= 2F
So, if qA is doubled, the new force will be twice the original force.
(b) What new force will exist if qA and qB are cut in half?
If both qA and qB are cut in half, we can calculate the new force as follows:
F_new = k * ((qA/2) * (qB/2)) / d^2
= (1/4) * (k * (qA * qB) / d^2)
= (1/4) * F
Therefore, if qA and qB are both cut in half, the new force will be one-fourth (1/4) of the original force.
(c) What new force will exist if d is tripled?
When the distance d between the charges is tripled, we can determine the new force as:
F_new = k * (qA * qB) / (3d)^2
= (1/9) * (k * (qA * qB) / d^2)
= (1/9) * F
So, if d is tripled, the new force will be one-ninth (1/9) of the original force.
(d) What new force will exist if d is cut in half?
If we halve the distance between the charges, the new force can be calculated as:
F_new = k * (qA * qB) / (d/2)^2
= 4 * (k * (qA * qB) / d^2)
= 4F
So, if d is cut in half, the new force will be four times the original force.
(e) What new force will exist if qA is tripled and d is doubled?
To find the new force when qA is tripled and d is doubled, we can substitute the modified values into the Coulomb's law formula:
F_new = k * (3qA * qB) / (2d)^2
= (9/4) * (k * (qA * qB) / d^2)
= (9/4) * F
Therefore, if qA is tripled and d is doubled, the new force will be (9/4) times the original force.
By analyzing Coulomb's law and applying the appropriate modifications to the variables, we can determine the new forces in each scenario.