Mastering Physics: Exercise 23.5

Mastering Physics: Exercise 23.5

A small metal sphere, carrying a net charge of q1 = -2.70 μC , is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of q2 = -7.50 μC and mass 1.70 g , is projected toward q1. When the two spheres are 0.800 m apart, q2 is moving toward q1 with speed 22.0 m/s (Figure 1) . Assume that the two spheres can be treated as point charges. You can ignore the force of gravity.

Figure 1

Part A

Question: What is the speed of q2 when the spheres are 0.420 m apart?

Answer: 

K1 + U1 = K2 + U2

K1 = 1/2 × mv^2 = 1/2 × 1.70 × 10^(−3) kg × (22)^2 = 0.4114

U1 = 1/(4πrϵ_0) × (q × q2) / r 

U1 =  8.99 × 10^(9) × −2.70 × 10^(−6) × −7.50 × 10^(−6) / (0.806)

U1 = 0.227559

U2 = 1/(4πrϵ_0) × (q × q2) / r 

U2 =  8.99 × 10^(9) × −2.70 × 10^(−6) × −7.50 × 10^(−6) / (0.420)

U2 = 0.433446

=> K2 = K1 +U1 − U2

K2 = 0.4114 + 0.227559 − 0.433446

K2 = 0.205513

1/2 × mv^2  = 0.205513

=> v =  15.5 m/s 

Part B

Question: How close does q2 get to q1?

Answer: 

K2 = 0

K1 + U1 = U2 + 0

K1 + U1 = 0.4114 + 0.227559

K1 + U1 = 0.638959

U2 = 0.638959

1/(4πrϵ_0) × (q × q2) / r = 0.638959

=> r = −2.70 × 10^(−6) × −7.50 × 10^(−6) / (0.638959 / (8.99 × 10^(9))

=> r =  0.285 m