Question

0.16 Write and prove "Gauss' theorem" for uniform spherical shell.
Q.16 गौस का प्रमेय लिखिए तथा सममित गोलीयकोष हेतु इसकारात्यापन कीजिए।
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Answer 1

INTRODUCTION

Gauss's law states that any charge qq can be thought to give rise to a definite quantity of flux through any enclosing surface. Physically, we might think of any source of light, such as a lightbulb, or the Sun, which has a definite rating of power which it emits in all directions. No matter what shape of enclosing surface we trap it in, and no matter how near or far the light source is from the surface, the enclosing surface will receive the same amount of energy per unit time P_0P

In a similar way, a distribution of total charge \sum_i q_i will give rise to an invariant amount of flux through any enclosing surface, which is defined to be the sum over all infinitesimal patches of surface dAdA, of the electric field component perpendicular to the surface. Although not always a practical tool, in situations where geometrical symmetries can be exploited, Gauss's law is an incredibly powerful tool to quickly calculate electric fields. Analogous laws hold for other inverse square laws, e.g. Newtonian gravity.

PROOF

We start with Coulomb's law for a single point charge E = \frac{\gamma}{r^2}, where \gamma = \frac{q}{4\pi \varepsilon_0}

Now, we integrate the electric field over a closed spherical surface encasing the charge

∮E.n^ dA=∮\frac{\gamma}{r^2} dA

=\frac{\gamma}{r^2} ∮dA

=\frac{\gamma}{r^2} 4\pi r^2

=\frac{q}{\varepsilon_0}

APPLICATIONS

Gauss's law is a powerful statement about inverse square fields. Not just in problem-solving, it has also found its place in the four Maxwell equations as well as in gravity.

Gauss’s Law can be used to solve complex electrostatic problems involving unique symmetries like cylindrical, spherical or planar symmetry. Also, there are some cases in which calculation of electric field is quite complex and involves tough integration. Gauss’s Law can be used to simplify evaluation of electric field in a simple way.

• Choose a Gaussian surface, such that evaluation of electric field becomes easy
• Make use of symmetry to make problems easier
• Remember, it is not necessary that Gaussian surface to coincide with real surface that is, it can be inside or outside the Gaussian surface

Hope this helps