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10 votes
10 votes
In the electric field
{\vec{E}=3x\hat{i}-2y\hat{j}+5z\hat{k}}, find the potential difference between the points A(1,3,5) and B(3,2,7)​

User Omer YILMAZ
by
3.0k points

1 Answer

19 votes
19 votes

Let's recall that, the potential difference between any two points X(x,y,z) and Y(a,b,c) is given by ;


  • {\boxed{\bf{V_(Y)-V_(X)=\displaystyle \bf -\int_(X)^(Y)\overrightarrow{E}\cdot \overrightarrow{dr}}}}

So, here ;


{:\implies \quad \sf \overrightarrow{E}=3x\hat{i}-2y\hat{j}+5z\hat{k}}

So, now our second component of the Integrand will just be ;


{:\implies \quad \sf \overrightarrow{dr}=dx\hat{i}+dy\hat{j}+dz\hat{k}}

So, now the whole integrand will just be ;


{:\implies \quad \sf \overrightarrow{E}\cdot \overrightarrow{dr}=(3x\hat{i}-2y\hat{j}+5z\hat{k})(dx\hat{i}+dy\hat{j}+dz\hat{k})}


{:\implies \quad \sf \overrightarrow{E}\cdot \overrightarrow{dr}=3xdx-2ydy+5zdz}

Now, Let's move to the final answer ;


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\int_(X)^(Y)3xdx-2ydy+5zdz}

As,X is the point (1,3,5) and Y being (3,2,7) , so seperate the integral into three integrals with limits as follows respectively;


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\bigg(\int_(1)^(3)3xdx-\int_(3)^(2)ydy+\int_(5)^(7)zdz\bigg)}


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\bigg(3\int_(1)^(3)xdx-2\int_(3)^(2)ydy+5\int_(5)^(7)zdz\bigg)}


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\bigg\_(3)^(2)+5\bigg((z^2)/(2)\bigg)\bigg}


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\bigg\{2\bigg((9)/(2)-\frac12\bigg)-2\bigg((4)/(2)-\frac92\bigg)+5\bigg((49)/(2)-(25)/(2)\bigg)\bigg\}}


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\{3(4)-(-5)+5(12)\}}


{:\implies \quad \displaystyle \sf V_(Y)-V_(X)=-\{12+5+60\}}


{:\implies \quad \displaystyle \boxed{\bf{V_(Y)-V_(X)=-77\:\: Volt}}}

Hence, this is the required answer

User Maryam Arshi
by
2.5k points
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