Answer :
Sure! Let's analyze the problem step by step using Coulomb's Law, which is given by the formula:
[tex]\[ F = k \frac{q_1 q_2}{d^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the force between the two charges,
- [tex]\( k \)[/tex] is Coulomb's constant,
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the charges, and
- [tex]\( d \)[/tex] is the distance between the charges.
### Original Situation:
The original force between the charges is denoted as [tex]\( F \)[/tex].
### New Situation:
1. Increasing [tex]\( q_1 \)[/tex] to twice its original value:
- The new charge [tex]\( q_1' \)[/tex] is [tex]\( 2q_1 \)[/tex].
2. Doubling the distance [tex]\( d \)[/tex]:
- The new distance [tex]\( d' \)[/tex] is [tex]\( 2d \)[/tex].
We need to find the new force [tex]\( F_{\text{new}} \)[/tex] with these changes.
Using Coulomb's Law for the new situation:
[tex]\[ F_{\text{new}} = k \frac{(2q_1) q_2}{(2d)^2} \][/tex]
Now, simplify the expression:
1. Substitute [tex]\( q_1' = 2q_1 \)[/tex] and [tex]\( d' = 2d \)[/tex]:
[tex]\[ F_{\text{new}} = k \frac{2q_1 q_2}{(2d)^2} \][/tex]
2. Calculate the square of the new distance ([tex]\( 2d \)[/tex]):
[tex]\[ (2d)^2 = 4d^2 \][/tex]
3. Substitute this back into the formula:
[tex]\[ F_{\text{new}} = k \frac{2q_1 q_2}{4d^2} \][/tex]
4. Simplify the fraction:
[tex]\[ F_{\text{new}} = k \frac{q_1 q_2}{2 \cdot 2d^2} = k \frac{q_1 q_2}{2 \cdot 2 \cdot d^2} = \frac{1}{2} \left( k \frac{q_1 q_2}{d^2} \right) \][/tex]
Remember, [tex]\( k \frac{q_1 q_2}{d^2} = F \)[/tex]:
[tex]\[ F_{\text{new}} = \frac{1}{2} F \][/tex]
Thus, the new force acting between the charges is [tex]\( \frac{1}{2} F \)[/tex].
### Conclusion:
The correct answer is:
[tex]\[ \boxed{\frac{1}{2}F} \][/tex]
[tex]\[ F = k \frac{q_1 q_2}{d^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the force between the two charges,
- [tex]\( k \)[/tex] is Coulomb's constant,
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the charges, and
- [tex]\( d \)[/tex] is the distance between the charges.
### Original Situation:
The original force between the charges is denoted as [tex]\( F \)[/tex].
### New Situation:
1. Increasing [tex]\( q_1 \)[/tex] to twice its original value:
- The new charge [tex]\( q_1' \)[/tex] is [tex]\( 2q_1 \)[/tex].
2. Doubling the distance [tex]\( d \)[/tex]:
- The new distance [tex]\( d' \)[/tex] is [tex]\( 2d \)[/tex].
We need to find the new force [tex]\( F_{\text{new}} \)[/tex] with these changes.
Using Coulomb's Law for the new situation:
[tex]\[ F_{\text{new}} = k \frac{(2q_1) q_2}{(2d)^2} \][/tex]
Now, simplify the expression:
1. Substitute [tex]\( q_1' = 2q_1 \)[/tex] and [tex]\( d' = 2d \)[/tex]:
[tex]\[ F_{\text{new}} = k \frac{2q_1 q_2}{(2d)^2} \][/tex]
2. Calculate the square of the new distance ([tex]\( 2d \)[/tex]):
[tex]\[ (2d)^2 = 4d^2 \][/tex]
3. Substitute this back into the formula:
[tex]\[ F_{\text{new}} = k \frac{2q_1 q_2}{4d^2} \][/tex]
4. Simplify the fraction:
[tex]\[ F_{\text{new}} = k \frac{q_1 q_2}{2 \cdot 2d^2} = k \frac{q_1 q_2}{2 \cdot 2 \cdot d^2} = \frac{1}{2} \left( k \frac{q_1 q_2}{d^2} \right) \][/tex]
Remember, [tex]\( k \frac{q_1 q_2}{d^2} = F \)[/tex]:
[tex]\[ F_{\text{new}} = \frac{1}{2} F \][/tex]
Thus, the new force acting between the charges is [tex]\( \frac{1}{2} F \)[/tex].
### Conclusion:
The correct answer is:
[tex]\[ \boxed{\frac{1}{2}F} \][/tex]
