using gauss law derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical - Physics - Electric Charges And Fields

Question

using gauss law derive an expression for the electric field
intensity at any point outside a uniformly charged thin spherical
shell of radius R and charge density sigma coloumb per metre square
draw the field lines when the charge density of the sphere is
1)positive 2)negative

Ruhhi Ralhan · Accepted Answer

The charged spherical shell has surface density σ C/m² which can be written like

$σ = \frac{Total charge on the spherical shell}{Spherical area of the shell} = \frac{q}{4 π R^{2}} o r, q = 4 σ π R^{2} .................. (1)$

Draw the Gaussian surface through point P which will be a spherical shell of radius r encloses the charged spherical shell.

Let Vector E be the electric field at a point on gaussian shell. Consider a small area element vector dS around the point. Then the electric flux through area element,

$d Φ = \vec{E} . \vec{d S}$

As vector dS is normal along the surface of shell, it is along the direction of vector E. Thus,

$d Φ = E d S Total electric flux, Φ = \oint_{S} E d S = E \oint_{S} d S Now, \oint_{S} d S = Surface area of spherical shell of radius r=4 π r^{2} ∴ Φ = E \times 4 π r^{2} ............... (1)$

According to Gauss theorem,

$Φ = \frac{q}{ϵ_{0}} ................. (2) Form eq(1) and eq(2) E \times 4 π r^{2} = \frac{q}{ϵ_{0}} o r, E = \frac{q}{4 π ϵ_{0} r^{2}} For (r>R)$

Electric field lines due to a positive spherical shell:

using gauss law derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and charge density sigma coloumb per metre square. draw the field lines when the charge density of the sphere is 1)positive 2)negative