Question

Point charge (q) moves form point (P) to point (S) along the path PQRS as shown in fig. in a uniform electric field E. Pointing along to the positive direction of the x-axis. The co-ordinates of the points P,Q,R and S are (a, b, 0), (2a, 0, 0), (a, –b, 0) and (0, 0, 0) respectively. The work done by the field in the above process is given by the expression - (A) qEa (B) -qEa (C) qEA√2 (D) qE$\sf{\sqrt{\left(2a\right)^2+b^2}}$

Answer 1

Answer:

• The Work done is - qEa Joules

Explanation:

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We know that work done by a conservative force is path independent. Therefore, work done in path PQRS will be equal to the work done in PS path.

Now, lets Find the Displacement SP,

$\longrightarrow\sf \overrightarrow{\sf SP} =r_{2}-r_{1}\\\\\\\longrightarrow\sf \overrightarrow{\sf SP} = \bigg(0\;\hat{i}+0\;\hat{j}+0\;\hat{k}\bigg)-\bigg(a\;\hat{i}+b\;\hat{j}+0\;\hat{k}\bigg)\\\\\\\longrightarrow\sf \overrightarrow{\sf SP}=-a\;\hat{i}-b\;\hat{j}$

∴ We got displacement vector.

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Now, from the formula we know,

$\large\bigstar\;\underline{\boxed{\sf W=q\;V}}$

Here,

• W Denotes Work done.
• q Denotes Charge.
• V Denotes Potential.

Now,

$\longrightarrow\sf W=qV\\\\\\\longrightarrow\sf W=q\times \bigg(\overrightarrow{\sf E}\;.\;\overrightarrow{\sf ds}\bigg)\ \ \ \because\Bigg[V=\overrightarrow{\sf E}\;.\;\overrightarrow{\sf ds}\Bigg]$

Now let's write it through Cartesian system.

$\longrightarrow\sf W=q\times \bigg(\overrightarrow{\sf E_{x}}\;.\;\overrightarrow{\sf dx}+\overrightarrow{\sf E_{y}}\;.\;\overrightarrow{\sf dy}+\overrightarrow{\sf E_{z}}\;.\;\overrightarrow{\sf dz}\bigg)$

Substituting the values,

But as we know that electric field is along x - axis. so electric field along y and z axis is 0.So,

$\longrightarrow\sf W=q\times \bigg(E\;\hat{i}\times -a\;\hat{i}+0\;\hat{j}\times -b\;\hat{j}\bigg)\\\\\\\longrightarrow\sf W=q\times \bigg(E\;\hat{i}\times -a\;\hat{i}\bigg)\\\\\\\longrightarrow\sf W=q\times -Ea\\\\\\\longrightarrow\sf W=-\;qEa\\\\\\\longrightarrow \large{\underline{\boxed{\red{\sf W=-\;qEa\;J}}}}$

∴ The Work done is - qEa Joules.

Hence Option - B is correct!

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Answer 2

Explanation:

• the work done is -qEa Joules