A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge +q. If it strikes the surface of sphere with speed u, find the minimum speed u, so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces.)
(a) \(\frac{q}{\sqrt{2\pi \epsilon_0 mR}}\)
(b) \(\frac{q}{\sqrt{4\pi \epsilon_0 mR}}\)
(c) \(\frac{q}{\sqrt{8\pi \epsilon_0 mR}}\)
(d) \(\frac{\sqrt{3}q}{\sqrt{4\pi \epsilon_0 mR}}\)
Answer (1)
To solve this problem, we need to consider the electrostatic forces and energy conservation as the bullet, which is charged, interacts with the charged sphere.
Initially, the bullet is at a distance where electrostatic potential energy with respect to the sphere is negligible compared to when it's at the surface. As it approaches the sphere, it must overcome the electrostatic potential energy barrier due to the sphere's charge.
Key Concepts:
Electrostatic Potential Energy : When the bullet reaches the surface of the sphere, its potential energy due to the electrostatic force can be calculated using Coulomb's Law. For a sphere with a uniform charge distribution, the electric potential at the surface can be given as:
V = R k ⋅ q = 4 π ϵ 0 1 ⋅ R q
where:
k = 4 π ϵ 0 1 is the electrostatic constant.
q is the charge of the bullet and the sphere.
R is the radius of the sphere.
Conservation of Energy : The initial kinetic energy of the bullet must be equal to the electrostatic potential energy plus the kinetic energy needed to just enter the sphere:
2 1 m u 2 = 4 π ϵ 0 1 ⋅ R q 2
Rearranging for u :
u = 2 π ϵ 0 m R q 2
With these calculations, the correct minimum speed u of the bullet such that it can just penetrate the sphere is:
Option (a): 2 π ϵ 0 m R q
This detailed solution explains how to calculate the minimum speed using the principles of electrostatic potential and energy conservation.
