The displacement of a charge Q in the electri | vedclass

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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 dot product formula:$W = Q(\v

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$Q(ae_1 + be_2)$

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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 dot product formula:$W = Q(\vec{E} \cdot \vec{r})$Given $\vec{E} = e_1\hat{i} + e_2\hat{j} + e_3\hat{k}$ and $\vec{r} = a\hat{i} + b\hat{j}$.Substituting these values into the formula:$W = Q[(e_1\hat{i} + e_2\hat{j} + e_3\hat{k}) \cdot (a\hat{i} + b\hat{j})]$Using the properties of dot products ($\hat{i} \cdot \hat{i} = 1, \hat{j} \cdot \hat{j} = 1, \hat{i} \cdot \hat{j} = 0$,etc.):$W = Q(e_1 \cdot a + e_2 \cdot 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 is

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