A block of mass m and positive charge q is placed on an insulated frictionless inclined plane as shown in the figure. A uniform electric field E is applied parallel to the inclined surface such that the block is at rest. Calculate the magnitude of the electric field E.

Explanation

There are three forces that acts on the mass m:
(i) The downward gravitational force exerted by the Earth (mg)
(ii) The normal force exerted by the inclined surface (N)
(iii) The Coulomb force given by uniform electric field (qE)
The free body diagram for the mass ma is drawn below . A convenient inertial of the mass m is drawn below. A convenient inertial coordinate system is located in the inclined surface as shown in the figure. The mass m has zero net acceleration both in x and y -direction. Along x-direction applying Newton.s second law we have
`mg sin thetahati-qEhati=0`
`mg sin theta-qEhati=0`
or ,` E=(mg sin theta)/(q)`

Note that the magnitude of the electric field is drectly proportional to the mass m and inversely proportional to the charge q . It implies that if the mass is increased by keeping the charge constant then a strong electric field is required to stop the object from sliding . If the charge is increaed by keeping the mass constant then a weak electric field is sufficient to stop the mass from sliding down the plane . The electric field also can be expressed in terms of height and the length of the inclined surface of the plane
`E=(mgh)/(qL)`