Question

this is a question of physics class 12th chapter 1 electric charges and fields. 1) find electric field due to long wire having uniform linear charge density by using Gaus's law ​

Answer 1

Explanation:

Consider a point A at a perpendicular distance l from the mid-point O of the wire, as shown in the following figure.

Let E be the electric field at point A due to the wire, XY.

Consider a small length element dx on the wire section with OZ = x

Let q be the charge on this piece.

∴ q = λdx

Electric field due to the piece,

However,

The electric field is resolved into two rectangular components. dEcosθ is the perpendicular component and dEsinθ is the parallel component. When the whole wire is considered, the component dEsinθ is cancelled. Only the perpendicular component dEcosθ affects point A.

Hence, effective electric field at point A due to the element dx is dE1 .

∴ d E1 = 1/4πε0 x λdx. cos θ/(l2 + x2) ...(1)

In ∆AZO, tanθ = x/l ⇒ x = l.tanθ ...(2)

On differentiating equation (2), we obtain

dx/dθ = l x sec 2 θ ⇒ dx = l x sec 2 θdθ ...(3)

From equation (2), we have

x2 + l2 = l2tan2θ + l2 = l2 (tan2 θ + 1) = l2 sec2 θ ...(4)

Putting equations (3) and (4) in equation (1), we obtain

The wire is so long that θ tends from − π/2 to π/2.

By integrating equation (5), we obtain the value of field E1 as,

Therefore, the electric field due to long wire is λ/2πε0 l

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