The displacement of a charge Q in the electri | vedclass

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Question

(A) The work done $W$ by an electric field $\vec{E}$ on a charge $Q$ undergoing a displacement $\vec{r}$ is given by the formula: $W = Q (\vec{E} \cdo

Vedclass · Accepted Answer

$Q (ae_1 + be_2)$

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(A) The work done $W$ by an electric field $\vec{E}$ on a charge $Q$ undergoing a displacement $\vec{r}$ is given by the formula: $W = Q (\vec{E} \cdot \vec{r})$ Given $\vec{E} = e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}$ and $\vec{r} = a \hat{i} + b \hat{j} + 0 \hat{k}$. Substituting these values into the dot product: $W = Q [(e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}) \cdot (a \hat{i} + b \hat{j} + 0 \hat{k})]$ Using the property of dot product $\hat{i} \cdot \hat{i} = 1$,$\hat{j} \cdot \hat{j} = 1$,$\hat{k} \cdot \hat{k} = 1$ and cross terms are $0$: $W = Q (e_1 a + e_2 b + e_3 \cdot 0)$ $W = Q (ae_1 + be_2)$

The displacement of a charge $Q$ in the electric field $\vec{E} = e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}$ is $\vec{r} = a \hat{i} + b \hat{j}$. The work done by the electric field is:

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