Question

A nonconducting sphere of diameter 10.0 cm carries charge distributed uniformly inside with charge density of +5.50 µC/m3 . A proton is released from rest just outside the surface. Will it move away from or move closer to the sphere? How far will it be from the SURFACE of the sphere when its speed has reached 2550 km/s? (k = 1/4πε0 = , , [3+7]

Answer 1

t = 2.58*10^-6 s

Step-by-step explanation:

For a nonconducting sphere you have that the value of the electric field, depends of the region:

`r<R:\\\\E=k(Qr)/(R^3)\\\\r>R:\\\\E=k(Q)/(r^2)`

k: Coulomb's constant = 8.98*10^9 Nm^2/C^2

R: radius of the sphere = 10.0/2 = 5.0cm=0.005m

In this case you can assume that the proton is in the region for r > R. Furthermore you use the secon Newton law in order to find the acceleration of the proton produced by the force:

`F=m_pa\\\\qE=m_pa\\\\k(qQ)/(r^2)=m_pa\\\\a=k(qQ)/(m_pr^2)`

Due to the proton is just outside the surface you can use r=R and calculate the acceleration. Also, you take into account the charge density of the sphere in order to compute the total charge:

`Q=\rho V=(5.5*10^(-6)C/m^3)((4)/(3)\pi(0.05m)^3)=2.87*10^(-9)C\\\\a=(8.98*10^9Nm^2/C^2)((1.6*10^(-19)C)(2.87*10^(-9)C))/((1.67*10^(-27)kg)(0.05m)^2)=9.87*10^(11)(m)/(s^2)`

with this values of a you can use the following formula:

`a=(v-v_o)/(t)\\\\t=(v-v_o)/(a)=(2550*10^3m/s-0m/s)/(9.87*10^(11)m/s^2)=2.58*10^(-6)s`

hence, the time that the proton takes to reach a speed of 2550km is 2.58*10^-6 s