Two particles each have the same mass but particle #1 has four times the charge of particle #2. Particle #1 is accelerated from rest through a potential difference of 10 V and attains speed v. Particle #2 is accelerated from rest also through a potential difference of 10 V. What speed does particle #2 attain?

 v_2 = 2*v  

Explanation:

Given:

- Mass of both charges = m

- Charge 1 = Q_1

- Speed of particle 1 = v

- Charge 2 = 4*Q_1

- Potential difference p.d = 10 V

Find:

What speed does particle #2 attain?

Solution:

- The force on a charged particle in an electric field is given by:

                                       F = Q*V / r

Where, r is the distance from one end to another.

- The Net force acting on a charge accelerates it according to the Newton's second equation of motion:

                                      F_net = m*a

- Equate the two expressions:

                                      a = Q*V / m*r

- The speed of the particle in an electric field is given by third kinetic equation of motion.

                                      v_f^2 - v_i^2 = 2*a*r

Where, v_f is the final velocity,

            v_i is the initial velocity = 0

                                      v_f^2 - 0 = 2*a*r

Substitute the expression for acceleration in equation of motion:

                                       v_f^2 = 2*(Q*V / m*r)*r

                                       v_f^2 = 2*Q*V / m

                                       v_f = sqrt (2*Q*V / m)

- The velocity of first particle is v:

                                       v = sqrt (20*Q / m)

- The velocity of second particle Q = 4Q

                                       v_2 = sqrt (20*4*Q / m)

                                       v_2 = 2*sqrt (20*Q / m)

                                       v_2 = 2*v