The displacement of a charge Q in the electric field E^-=e1 i ∧+e2 j ∧+e3 k ∧is r^-=a i ∧+b j ∧T

step 1 Hi friends here we have to calculate the work done in displacing the charge Q charge Q in electr

The displacement of a charge Q in the electric field E^-=e1...

The displacement of a charge $Q$ in the electric field $E^{-}=e_{1} i \wedge+e_{2} j \wedge+e_{3} k \wedge$ is $r^{-}=a i \wedge+b j \wedge$ The work done is $\ldots \ldots$ (A) $Q\left(e_{1}+e_{2}\right) \sqrt{\left(a^{2}+b^{2}\right)}$ (B) $Q\left[\sqrt{ \left.\left(e_{1}^{2}+e_{2}^{2}\right)\right](a+b)}\right.$ (C) $Q\left(a e_{1}+b e_{2}\right)$ (D) $\left.Q \sqrt{[}\left(a e_{1}\right)^{2}+\left(b e_{2}\right)^{2}\right]$

The displacement of a charge $Q$ in the electric field $E^{-}=e_{1} i \wedge+e_{2} j \wedge+e_{3} k \wedge$ is $r^{-}=a i \wedge+b j \wedge$
The work done is $\ldots \ldots$
(A) $Q\left(e_{1}+e_{2}\right) \sqrt{\left(a^{2}+b^{2}\right)}$
(B) $Q\left[\sqrt{ \left.\left(e_{1}^{2}+e_{2}^{2}\right)\right](a+b)}\right.$
(C) $Q\left(a e_{1}+b e_{2}\right)$
(D) $\left.Q \sqrt{[}\left(a e_{1}\right)^{2}+\left(b e_{2}\right)^{2}\right]$

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Key Concepts

Electric Field

An electric field is a vector field that represents the force per unit charge that would be exerted on a small positive test charge at any point in space. This concept is essential in understanding how charges interact in space and form the basis for calculating forces and potentials in electrostatics.

Dot Product

The dot product is a fundamental operation in vector calculus that multiplies two vectors to yield a scalar. It is used to find the magnitude of the projection of one vector onto another, which is particularly useful in physics for calculating work, where only the component of force in the direction of displacement contributes to the work done.

Work Done in a Vector Field

Work is calculated as the dot product of the force vector and the displacement vector. In the context of an electric field, when a charge moves in the presence of a force, the work done by the field on the charge is determined by summing the products of the components of the force and the corresponding components of the displacement. This concept links the geometry of vector fields to the physical work performed on an object.

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