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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=λ2πϵ0a2×12(^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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