. A PPLICATIONS OF G AUSS ’ S L AW The electric field due to a general charge distribution is, as seen above, given by Eq. ( . ).
In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space. For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law. This is best understood by some examples. .
. Field due to an infinitely long straight uniformly charged wire Consider an infinitely long thin straight wire with uniform linear charge density λ . The wire is obviously an axis of symmetry. Suppose we take the radial vector from O to P and rotate it around the wire.
The points P, P ′ , P ′′ so obtained are completely equivalent with respect to the charged wire. This implies that the electric field must have the same magnitude at these points. The direction of electric field at every point must be radial (outward if λ > , inward if λ < ). This is clear from Fig.
. . Consider a pair of line elements P and P of the wire, as shown. The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel).
This is true for any such pair and hence the total field at any point P is radial. Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire. In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distance r . To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig.
. (b). Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero. At the cylindrical part of the surface, E is normal to the surface