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A thin wire carrying current i is bent t...

A thin wire carrying current i is bent to form a closed loop of one turn. The loop is placed in y-z plane with centre at origin. If R is the radius of the loop, then

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The electric field vectors due to the threadat points p and Q on the ring is

`vec(E_(p))= (lambda)/(4pi epsilon_(0)R)[-hat(i)+hat(j)]`
`vec(E_(Q))= (lambda)/(4pi epsilon_(0)R) [-hat(i)-hat(j)]`
The force experienced by small charge elements dq at P ad Q is
`d vec(F_(P))=(lambdadq)/(4piepsilon_(0)R)[-hat(i)+hat(j)],dvec(F_(Q))= (lambdadq)/(4piepsilon_(0)R)[-hat(i)-hat(j)]`
Clearly, the components of these forces add up along -x -xis and cancel out alont y-axis . Thus, total force experienced by the ring is along -x-axis. .
`implies` Net force , `F_(x) = int dF_(x)`
`implies F_(x) = - int ( lambdadq)/(4piepsilon_(0)R)= (- lambdaQ)/(4piepsilon_(0)R)`
or `vec(F) = ( lambdaQ vec(i))/(4pi epsilon_(0)R)`
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