A) \[-(3\hat{i}+5\hat{j}+3\hat{k})\]
B) \[-(6\hat{i}+5\hat{j}+2\hat{k})\]
C) \[-(2\hat{i}+3\hat{j}+\hat{k})\]
D) \[-(6\hat{i}+9\hat{j}+\hat{k})\]
Correct Answer: B
Solution :
Given, potential in a region, \[V=6xy-y+2yz.\] Electric field in a region, \[E=-\frac{\partial V}{\partial x}\mathbf{\hat{i}}-\frac{\partial V}{\partial y}\mathbf{\hat{j}}-\frac{\partial V}{\partial z}\mathbf{\hat{k}}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E=-6y\mathbf{\hat{i}}-(6x-1)\mathbf{\hat{j}}-2y\mathbf{\hat{k}}\] At, (1, 1, 0), electric field can be expressed, \[E=-(6\times 1\mathbf{\hat{i}})-(6\times 1-1)\mathbf{\hat{j}}-2\times 1.\mathbf{\hat{k}}\] \[=-(6\mathbf{\hat{i}}+5\mathbf{\hat{j}}+2\mathbf{\hat{k}})N/C\]
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