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Saitama
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Homework Statement
Electric field given by the vector [itex]\vec{E}=x\hat{i}+y\hat{j}[/itex] is present in xy plane. A small ring carrying charge +Q, which can freely slide on a smooth non conducting rod, is projected along the rod from point (0,L) such that it can reach the other end of the rod. What minimum velocity should be given to the rod? (Assume zero gravity).
Homework Equations
The Attempt at a Solution
I tried finding the change in potential, finding the work done and equating the work done to kinetic energy. Let's assume that it moves a very small displacement dr as shown in the figure.
It will have components dx[itex]\hat{i}[/itex] and -dy[itex]\hat{j}[/itex].
∴[itex]dr=dx\hat{i}-dy\hat{j}[/itex].
Change in potential, ΔV
[tex]ΔV=-\int\vec{E}\cdot\vec{dr}[/tex]
[tex]=-\int(x\hat{i}+y\hat{j})\cdot(dx\hat{i}-dy\hat{j})[/tex]
[tex]=-\int_{0}^{L}xdx+\int_L^0ydy[/tex]
[tex]=-L^2[/tex]
[tex]QΔV=\frac{1}{2}mu^2[/tex]
where u is the velocity with which ring is projected.
Solving this i get,
[tex]u=\sqrt{\frac{2QL^2}{m}}[/tex]
But this is wrong, the answer is
[tex]u=\sqrt{\frac{QL^2}{2m}}[/tex]
I don't understand where i am doing wrong.
Any help is appreciated.