Sphere 1 with radius has positive charge . Sphere 2 with radius is far from sphere 1 and initially uncharged. After the separated spheres are connected with a wire thin enough to retain only negligible charge, (a) is potential of sphere 1 greater than, less than, or equal to potential of sphere 2? What fraction of ends up on (b) sphere 1 and (c) sphere 2? (d) What is the ratio of the surface charge densities of the spheres?
The radius of sphere 1 s .
The radius of sphere 2 is .
The electric potential due to the point charge is,
Here, is the coulomb’s constant, is the charge, and is the distance from the charge.
The potential of sphere 1 is,
Here, is the charge on sphere 1.
The potential of sphere 2 is,
Here, is the charge on sphere 2.
The total electric charge is,
The charges must be the same for both spheres when both spheres are connected by a thin wire. The electric potential is directly proportional to the charge. Hence, the electric potential on both spheres must be the same.
Here, V1 is the electric potential on sphere 1 and V2 is the electric potential on sphere 2. Hence, the electric potential of sphere 1 is equal to the electric potential on sphere 2.
The relation between the radius of the sphere 1 and sphere 2 is,
The relation between the potential of the sphere 1 and sphere 2 is,
Substitute for in the equation (3) and solve for .
Therefore, the charge on the sphere 1 in the fraction of is .
Substitute for in the equation (4) and solve for .
Hence, the charge on the sphere 2 in the fraction of is .
The surface charge density of sphere 1 is,
The surface charge density of sphere 2 is,
Take the ratio of the surface charge densities of sphere 1 and 2.
Substitute for and for in the above equation.
Hence, the ratio of the surface charge densities of the spheres is .
Question: An electron is placed in an x-y plane where the electric potential depends on x and y as shown, for the coordinate axes, in Fig. 24-51 (the potential does not depend on z). The scale of the vertical axis is set by Vs =500 V. In unit-vector notation, what is the electric force on the electron?
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