A charge "Q" is divided into two parts such that the electrostatic ...

Question

A charge "Q" is divided into two parts such that the electrostatic force between two parts is maximum. Find the charge on each part.

Accepted Answer

Setup:
Let one part be q and the other part be (Q - q).

Step 1: Express the force
Using Coulomb's law (with fixed distance r):
F = k · q · (Q - q)/r²
F = (k/r²) · (qQ - q²)

Step 2: For maximum force, differentiate F w.r.t. q
dF/dq = (k/r²) · (Q - 2q)

Step 3: Set derivative = 0
Q - 2q = 0
q = Q/2

Step 4: Verify it's a maximum
d²F/dq² = -2k/r² < 0 ✓ (Maximum confirmed)

Step 5: Charges on each part
First part = q = Q/2
Second part = Q - q = Q - Q/2 = Q/2

Conclusion:
The charge must be divided into two equal halves (Q/2 and Q/2) for maximum force.

Maximum force:
F_max = k · (Q/2) · (Q/2)/r² = kQ²/(4r²)

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