Let’s solve this step-by-step.
### Step 1: Understand the problem
You have a total charge [tex]\( Q \)[/tex] that is split into two portions: one portion is [tex]\( q \)[/tex] and the other is [tex]\( Q - q \)[/tex]. These two charges [tex]\( q \)[/tex] and [tex]\( Q - q \)[/tex] are separated by a distance [tex]\( r \)[/tex].
### Step 2: Write Coulomb's Law
Coulomb's Law states that the electrostatic force [tex]\( F \)[/tex] between two point charges is given by:
[tex]\[ F = k \frac{q_1 q_2}{r^2} \][/tex]
where [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges, and [tex]\( r \)[/tex] is the separation between the charges.
In this scenario:
- [tex]\( q_1 = q \)[/tex]
- [tex]\( q_2 = Q - q \)[/tex]
### Step 3: Express the force in terms of [tex]\( q \)[/tex]
[tex]\[ F = k \frac{q (Q - q)}{r^2} \][/tex]
### Step 4: Simplify the equation
Since [tex]\( k \)[/tex] and [tex]\( r^2 \)[/tex] are constants for this problem, we can focus on the numerator:
[tex]\[ F \propto q(Q - q) \][/tex]
[tex]\[ F \propto qQ - q^2 \][/tex]
### Step 5: Find the maximum force
To maximize the force, we need to find the derivative of the expression [tex]\( qQ - q^2 \)[/tex] with respect to [tex]\( q \)[/tex] and set it to zero.
### Step 6: Differentiate with respect to [tex]\( q \)[/tex]
[tex]\[ \frac{d}{dq} (qQ - q^2) = Q - 2q \][/tex]
### Step 7: Set the derivative equal to zero to find the critical points
[tex]\[ Q - 2q = 0 \][/tex]
### Step 8: Solve for [tex]\( q \)[/tex]
[tex]\[ 2q = Q \][/tex]
[tex]\[ q = \frac{Q}{2} \][/tex]
### Conclusion
The charge [tex]\( q \)[/tex] that maximizes Coulomb repulsion between [tex]\( q \)[/tex] and [tex]\( Q - q \)[/tex] when separated by a distance [tex]\( r \)[/tex] is:
[tex]\[ q = \frac{Q}{2} \][/tex]
Hence, the correct relation is [tex]\( q = \frac{Q}{2} \)[/tex].
