Two identical small spheres each of mass 'm' and and identical charge which decreases as they approach each other with approach velocity (relative velocity) of 'c/√x' , c being a positive constant and x is the separation between the spheres . 'F' capital of in the figure is a constant external force. Neglect the gravitational force. The separation between the spheres at which the leakage rate of charge by spheres becomes infinity. {Given:(mc^2)/F}?

Question

Answer

Solution

Electric Force and Electric Field

The electric force between the two spheres is given by Coulomb’s law, F = kq1q2/r^2, where k is the Coulomb’s constant, q1 and q2 are the charges on the spheres, and r is the separation between them.

The electric field at a point P due to a charged sphere of radius R and charge q is given by E = kq/(R^2 + x^2)^3/2, where x is the distance of the point P from the center of the sphere.

The electric field at a point P due to two charged spheres of equal charge q and separation x is given by E = 2kq/(R^2 + x^2)^3/2, where R is the radius of each sphere.

Motion of the Spheres

The spheres are moving towards each other with an approach velocity of c/√x.

The force acting on each sphere is the net force due to the electric force between the spheres and the external force F.

The acceleration of each sphere is given by a = (F - kq^2/x^2)/m.

Leakage of Charge

The charge on each sphere decreases as they approach each other due to leakage of charge.

The leakage rate of charge is proportional to the electric field at the surface of the sphere.

The leakage rate of charge for a sphere of radius R and charge q is given by dQ/dt = -kE(R)^2, where E(R) is the electric field at the surface of the sphere.

Finding the Separation at which the Leakage Rate becomes Infinity

At some separation x, the leakage rate of charge becomes infinity, which means that the electric field at the surface of the sphere becomes infinite.

The electric field at the surface of each sphere is given by E(R) = 2kq/(R^2 + x^2)^3/2.

Equating the electric field to infinity and solving for x, we get x = R√2.

Substituting x = R√2 in the expression for acceleration, we get a = (mc^2)/F.

Final Answer

The separation between the spheres at which the leakage rate of charge by spheres becomes infinity is x = R√2.

The acceleration of each sphere at this separation is a = (mc^2)/F.

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