Question

A particle, having a charge +5 μc, is initially at rest at the point x = 30 cm on the x axis. The particle begins to move due to the presence of a charge q that is kept fixed at the origin. Find the kinetic energy of the particle at the instant it has moved 15 cm from its initial position if (a) q =+15μc and (b) q = -15μc 3

Answer 1

Gain in kinetic energy = Difference in potential energy

Potential Energy,

$E = k \frac{q_{1}q_{2}}{r_{12}}$

Where q₁ and q₂ are the charges

r₁₂ is the distance between the charges

k is Coulomb's constant ≈ 9×10⁹

Given :

q₁ = +5 μC

q₂ = ±15 μC

r₁₂ = 30

To find: K.E. at the instant q₁ moved 15cm from initial position.

Initial potential energy

$E = 9 * 10^{9} *\frac{5*10^{-6} * 15*10^{-6}}{30*10^{-2}} =2.25$

(a) Since both charges are positive, the charges repel. Hence the charge q₁ moves away by 15 cm.

Hence r₁₂ = 30+15 =45 cm

$E' = 9 * 10^{9} *\frac{5*10^{-6} * 15*10^{-6}}{45*10^{-2}} =1.5$

K.E. = E - E' = 2.25 - 1.5 = 0.75

(b) Since both charges are opposite, the charges attract. Hence the charge q₁ moves closer by 15 cm.

Hence r₁₂ = 30-15 =15 cm

$E' = 9 * 10^{9} *\frac{5*10^{-6} * 15*10^{-6}}{15*10^{-2}} =4.5$

K.E. = E' - E = 4.5 - 2.25 = 2.25

Answer 2

Given:

Initial charge = +5  μc

Initial position = 30 cm

Final position = 15 cm  + Initial position

To find:

• Kinetic energy when q = +15 μc
• Kinetic energy when q = -15 μc

Solution:

Kinetic energy = Inital potential energy - Potential energy

Potential energy = k * charge 1 * charge 2 / Distance between the points

Where,

k - 9 * 10^9

Solution 1:

Here,

Charge 1 = +5

Charge 2 = +15

Distance = 30

Hence, Initial potential energy = 9 * 10^9 * 5 * 10^-6 * 15 * 10^-6 / 30 * 10^-2

Solving,

We get,

Potential energy = 2.25

Now,

Distance = 45

Substituting,

Potential energy = 9 * 10^9 * 5 * 10^-6 * 15 * 10^-6 / 45 * 10^-2

Hence,

Potential energy = 1.5

Hence, Kinetic energy = 2.25 - 1.5

Kinetic energy = 0.75

Solution 2:

Here,

Distance = 15

Potential energy = 9 * 10^9 * 5 * 10^-6 * 15 * 10^-6 / 15 * 10^-2

Hence,

Potential energy = 4.5

Hence, Kinetic energy = 4.5 * 2.25

Kinetic energy = 2.25