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| Gauss' Theorem |
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| We have already learnt to find the electric field intensity due to a charged conductor using Coulomb's law. Gauss' theorem can also be used to calculate the electric field intensity provided there is a symmetry in the charge distribution. |
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| Can be considered as an alternative to Coulomb's law for expressing the relationship between electric charge and electric field. This theorem was formulated by a German mathematician Karl Friedrich Gauss. |
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| Gauss' theorem states that the total electric flux through any closed surface is proportional to the total electric charge inside the surface. |
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Total electric flux  q |
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| But we have already obtained the relation |
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| This means the electric flux is independent of the radius of the surface but only depends on the charge enclosed by the surface. |
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| Let case I - when Gaussian surface is spherical in shape, a positive charge q be placed at the centre of an imaginary spherical surface of radius R as shown in the figure. |
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| By symmetry, the field due to the charge +q is radial and E is perpendicular to the sphere and is directed along the normal to the surface. |
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| So the angle between the normal and the electric intensity is zero. |
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| That is, df = E dA cos0 |
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| Which is nothing but the mathematical representation of Gauss' theorem. |
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| The total electric flux is, |
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| where dW is the solid angle subtended by the area dA at the point. |
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| Which is nothing but Gauss' theorem |
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