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Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.
How much Charge is on the Outer surface of the Conductor? Give Some Examples
If one end of an insulated rod be charged, the charge is confined to that end only. But when any part of a conductor is charged, the charge distributes itself over the whole surface. No charge is found to exist in the inside of a solid conductor or on the inner surface of a hollow conductor. According to the property of a conductor, charge may flow easily through it. Like charges repel each other and try to move away from each other as far as possible. So, they distribute themselves on the outer surface of the conductor, where their mutual distance becomes maximum.
Under special circumstances, charges may reside on the inner surface of a hollow conductor. A charged body is kept inside a hollow sphere in such a way that it does not touch the hollow sphere. In this case, unlike charges are induced on the inner surface of the hollow sphere and like charges on the outer surface.
Without removing the charged body the hollow sphere is touched by hand. The free charges on the outer surface of the hollow sphere move to the earth but the bound charges on the inner surface of the sphere exist there. As long as the inducing charge remains inside the hollow sphere, the bound charge also remains on the inner surface. When the charged body is removed, charges shift to the outer surface of the sphere.
Electric Screen: Any arrangement, which can keep any space free from external electrical influences, is called an electric screen [Fig.]. Charge always resides on the outer surface of a conductor. Electric screens are based on this Property.
A gold-leaf electroscope enclosed in a wire-gauge cage C is placed on an insulated base A . Now if a charged body is brought near to or in contact with the cage, no effect is produced on the electroscope because charge resides on the outer surface of the cage, not inside. So the region enclosed by the cage is free from external electrical influences.
The space inside a closed metallic box is free from electrical influences for the same reason. The valves of a radio are placed in metallic cases to shield it from external electrical influences. Sensitive electrical instruments are always kept within electric screens.
Distribution Of Charge On The Surface Of A Conductor (Surface Density Of Charge)
Distribution of charge on a conducting surface: Although the charge on a conductor distributes itself all over the surface, it should not however be concluded that the distribution is always uniform all over the surface. The distribution of charge depends on the shape of the conductor. Greater the curvature at any point greater will be the accumulation of charge at that point. In Fig., distributions of charge on charged conductors
of different shapes are shown by dotted lines B. The boundary of each conductor is shown by the line A . The density of charge in each case is roughly represented by the distance of the dotted line B from the boundary line A of each conductor.
Surface density of charge: The surface density of charge at a point on a charged conductor is the amount of charge per unit area of the surface of the conductor surrounding the point. Surface density of charge is generally denoted by the symbol σ. If Q be the charge distributed uniformly over the surface of area A of a spherical conductor having radius r, the surface density of charge is given by,
σ = \(\frac{Q}{A}\) = \(\frac{Q}{4 \pi r^2}\) or, σ ∝ \(\frac{1}{r^2}\)
So, the surface density of charge reduces with the increase of radius of the object concerned and vice versa. Hence, at sharp bends or pointed portions of a conductor, surface density of charge will be greatest. So a conductor having different curva-tures at different points has different surface densities of charge at those points.
Unit of σ :
Dimension of σ: [σ] = L-2T|
Numerical Examples
Example 1.
A hollow spherical conductor of radius 2 cm is charged with 62.8 state. Determine the surface density of charge on the inner and outer surfaces of the conductor. If the sphere be a solid one what will be the values of the above quantities?
Solution:
No charge resides on the inner surface of a hollow conductor. So surface density of charge on the internal surface of the hollow sphere is zero.
Surface density of charge on the external surface of the sphere,
σ = \(\frac{Q}{4 \pi r^2}\) ; [Q = 62.8 statC; r = 2 cm]
= \(\frac{62.8}{4 \pi(2)^2}\) = \(\frac{62.8}{16 \pi}\) = 1.249 statC.cm-2
If the sphere be a solid one, it has no internal surface. The sur-face area of a hollow sphere and that of a solid sphere of the same radius are equal. So the surface density of charge on the external surface of the solid sphere will be the same as that of the hollow sphere.
Example 2.
27 drops of water, each of radius 3 mm and having equal charge are combined to form a large drop. Find the ratio of the surface density of charge on the large drop to that on each small drop.
Solution:
Suppose, charge on each small drop of water is q . So the charge in the combined drop will be 27 q .
In the first case, surface density of charge,
σ1 = \(\frac{q}{4 \pi r^2}\) = \(\frac{q}{4 \pi(0.3)^2}\) [here, r = 3 mm = 0.3 cm]
If R be the radius of the large drop, we have,
\(\frac{4}{3}\)πR3 = 27 × \(\frac{4}{3}\)π(0.3)3
or, R = 0.9 cm
In the second case,
σ2 = \(\frac{27 q}{4 \pi R^2}\) = \(\frac{27 q}{4 \pi(0.9)^2}\)
∴ \(\frac{\sigma_2}{\sigma_1}\) = \(\frac{27 q}{4 \pi(0.9)^2}\) × \(\frac{4 \pi(0.3)^2}{q}\) = \(\frac{27 \times 0.09}{0.81}\) = \(\frac{3}{1}\)
∴ σ2 : σ1 = 3 : 1
Example 3.
Spi A hollow spherical conductor of radius 2 cm is electri-fied with 20 state. Determine the surface density of charge on the external surface of the conductor. [HS 03]
Solution:
Surface density of charge of a spherical conductor,
σ = \(\frac{Q}{4 \pi r^2}\) [Q = charge on the surface of the sphere;
r = radius of the sphere]
Here, Q = 20 statC; r = 2 cm
∴ σ = \(\frac{20}{4 \pi(2)^2}\) = 0.398 statC cm-2