A wire loop formed by joining two semici...

A wire loop formed by joining two semicircular wires of radii `R_1` and `R_2` carries a current I as shown in figure. Find the magnetic field induction at the centre O.

Text Solution

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The magnetic field induction at O due to current through both the semi circular loops will be in the same direction which is perpendicular to the current loop, downwards. Thus total magnetic field induction at O is
`B=B_1+B_2=(mu_0)/(4pi)(piI)/(R_1)+(mu_0)/(4pi)(piI)/(R_2)`
`=(mu_0I)/(4)[(1)/(R_1)+(1)/(R_2)]`

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`(mu_(0)I)/(4)((1)/(R_(2))-(1)/(R_(1)))`

`(mu_(0)I)/(4)((1)/(R_(1))-(1)/(R_(2)))`

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`mu_(0)I((1)/(R_(1))-(1)/(R_(2)))`

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`(mu_0i)/4(1/(R^(1))-1/(R^(2)))`

`(mu_0i)/4(R^(2)-R^(2))`

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In the figure shown there are two semicircles of radii r_(1) and r_(2) in which a current i is flowing. The magnetic induction at the centre O will be

`(mu_(0)i)/r(r_(1)+r_(2))`

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`(mu_(0)i)/4((r_(1)+r_(2))/(r_(1)r_(2)))`

`(mu_(0)i)/4((r_(1)-r_(2))/(r_(1)r_(2)))`