Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they

Question

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is :

$(Take k = \frac{1}{4 π ε_{0}} and assume k Q^{2} > G m^{2} where G is the Gravitational constant)$ [2026]

Smart Achievers · Accepted Answer

(1)

$Using energy conservation$

$(2) (\frac{1}{2} m u^{2}) - \frac{G m^{2}}{4 r} + \frac{K Q^{2}}{4 r} = - \frac{G m^{2}}{2 r} + \frac{K Q^{2}}{2 r}$

$u = \sqrt{\frac{1}{4 m r} (K Q^{2} - G m^{2})}$

Solution

1 $\sqrt{\frac{k Q^{2}}{4 m R} (1 - \frac{G m^{2}}{k Q^{2}})}$

2 $\sqrt{\frac{k Q^{2}}{2 m R} (1 - \frac{G m^{2}}{2 k Q^{2}})}$

3 $\sqrt{\frac{k Q^{2}}{2 m R} (1 - \frac{G m^{2}}{k Q^{2}})}$

4 $\sqrt{\frac{k Q^{2}}{4 m R} (1 + \frac{G m^{2}}{k Q^{2}})}$

Ans.
(1)

$Using energy conservation$

$(2) (\frac{1}{2} m u^{2}) - \frac{G m^{2}}{4 r} + \frac{K Q^{2}}{4 r} = - \frac{G m^{2}}{2 r} + \frac{K Q^{2}}{2 r}$

$u = \sqrt{\frac{1}{4 m r} (K Q^{2} - G m^{2})}$

Want Us to Email you About Special Offers & Updates?

Solution

1 kQ24mR(1-Gm2kQ2)

2 kQ22mR(1-Gm22kQ2)

3 kQ22mR(1-Gm2kQ2)

4 kQ24mR(1+Gm2kQ2)