1. Show that the normal component of electrostatic field has a di… - Vidyadip

Question

Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by

$(\text{E}_2-\text{E}_1).\hat{\text{n}}=\frac{\sigma}{\in_0}$

where $\hat{\text{n}}$ is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of $\hat{\text{n}}$ is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is σ $\hat{\text{n}}$ /ε₀.

Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another.

[Hint: For (a), use Gauss’s law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.]

✓

Answer

Electric fielcl on one side of a charged body is E₁ and electric field on the other side of the same body is E₂. If infinite plane charged body has a uniform thickness, then electric field due to one surface of the charged body is given by,

$\vec{\text{E}}_1=-\frac{\sigma}{2\in_0}\hat{\text{n}} \dots\dots(1)$

Where,

$\hat{\text{n}}$ = Unit vector normal to the surface at a point

$\sigma$ = Surface charge density at that point

Electric field due to the other surface of the charged body,

$\vec{\text{E}}_2=-\frac{\sigma}{2\in_0}\hat{\text{n}} \dots\dots(2)$

Electric field at any point due to the two surfaces,

$\vec{\text{E}}_2-\vec{\text{E}}_1=\frac{\sigma}{2\in_0}\hat{\text{n}}+\frac{\sigma}{2\in_0}\hat{\text{n}}=\frac{\sigma}{\in_0}\hat{\text{n}}$

$(\vec{\text{E}}_2-\vec{\text{E}}_1).\hat{\text{n}}=\frac{\sigma}{\in_0} \dots\dots(3)$

Since inside a closed conductor, $\vec{\text{E}}_1=0,$

$\therefore\vec{\text{E}}=\vec{\text{E}}_2=-\frac{\sigma}{2\in_0}\hat{\text{n}}$

Therefore, the electric fielcl just outside the conductor is $\frac{\sigma}{\in_0}\hat{\text{n}}.$

When a charged particle is moved from one point to the other on a closed loop, the work done by the electrostatic field is zero. Hence, the tangential component of electrostatic field Is continuous from one side of a charged surface to the other.

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