Question

# Find the electric field vector at P(a, a, a) due to three infinitely long lines of charges along the x,y and z− axis, respectively. The charge density, i.e., charge per unit length of each wire is λ.

A
λ3πϵ0a(^i+^j+^k)
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B
λ2πϵ0a(^i+^j+^k)
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C
λ22πϵ0a(^i+^j+^k)
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D
2λπϵ0a(^i+^j+^k)
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Solution

## The correct option is B λ2πϵ0a(^i+^j+^k) Let us consider the electric field due to wire (3) only, →E3=E^u where, ^u=(^icos45∘+^jcos45∘) is the unit vector, →E=λ2πϵ0OP=λ2πϵ0(a2+a2)12 Substituting the value of E and ^u, we get →E3=λ2πϵ0(a2+a2)12(^icos45∘+^jcos45∘) ⇒→E3=λ2πϵ0a√2×1√2(^i+^j) ⇒→E3=λ4πϵ0a(^i+^j) Similarly, electric field due to wires (1) and (2), →E1=λ4πϵ0a(^j+^k) →E2=λ4πϵ0a(^i+^k) Thus, net electric field at P, →Enet=→E1+→E2+→E3 ∴→Enet=λ2πϵ0a(^i+^j+^k) Hence, option (b) is the correct answer.

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