Question

Two identical small metal spheres with q1 > 0 and |q1| > |q2| attract each other with a force of magnitude 72.1 mN when separated by a distance of 1.41 m . F21 F12 1.41 m q1 q2 r1 = 21 µm r2 = 21 µm The spheres are then brought together until they are touching, enabling the spheres to attain the same final charge q. q1 → q q2 → q ∆ q After the charges on the spheres have come to equilibrium, they spheres are separated so that they are again 1.41 m apart. F21 F12 1.41 m q q Now the spheres repel each other with a force of magnitude 21.63 mN. What is the final charge on the sphere on the right? The value of the Coulomb constant is 8.98755 × 109 N · m2 /C 2 . Answer in units of µC. 0What is the initial charge q1 on the first sphere? Answer in units of µC.

Answer 1

The value of the final charge on the sphere on the right is; 2.1855 × 10^(-6) C

The initial charge q1 on the sphere is;

-2.365 × 10^(-6) C or +6.736 × 10^(-6) C

Forces between charges

We are told that the spheres are identical and as such;

Final charge = (q1 + q2)/2

Where q1 and q2 are charges on both sphere's.

Now, formula for force between two charges is;

F = k*q1*q2/r²

Where;

k is coulombs constant = 9 × 10^(9) C

r is distance of separation

We are given;

F = 72.1 mN = 0.0721 N

r = 1.41 m

Thus;

(9 × 10^(9) × q1 * q2)/(1.41²) = -0.0721

Thus;

q1*q2 = (0.0721 × 1.41^(2))/(9 × 10^(9))

q1*q2 = -15.93 × 10^(-12) C² - - - (eq 1)

Formula for final force will be;

F_f = k*(q1 + q2)²/4r²

F_f = (9 × 10^(9) × (q1 + q2)²)/(4 × 1.41²)

Since they repel each other with a force of 21.63 mN = 0.02163 N, then;

(9 × 10^(9) × (q1 + q2)²)/(4 × 1.41²) = 0.02163

(9 × 10^(9) × (q1 + q2)²) = 0.02163 × 4 × 1.41²

(q1 + q2)² = (0.02163 × 4 × 1.41²)/9 × 10^(9)

(q1 + q2)² = 19.11 × 10^(-12) C

q1 + q2 = 4.371 × 10^(-6) C - - - (eq 2)

Making q1 the subject in eq 1 gives;

q1 = (-15.93 × 10^(-12))/q2

Putting that in eq 2 gives;

((-15.93 × 10^(-12))/q2) + q2 = 4.371 × 10^(-6)

Multiply through by q2 to get;

(q2)² - (4.371 × 10^(-6)) - (15.93 × 10^(-12)) = 0

Solving using quadratic equation calculator gives;

q2 = +6.736 × 10^(-6) C or -2.365 × 10^(-6) C

q1 = -2.365 × 10^(-6) C or +6.736 × 10^(-6) C

Final charge = (4.371 × 10^(-6))/2

Final charge = 2.1855 × 10^(-6) C

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