Two circular coils $$1$$ and $$2$$ are made from the same wire but the radius of the $$1st$$ coil is twice that of the $$2nd$$ coil. What is the ratio of potential difference applied across them so that the magnetic field at their centres is the same ?

Question

Infinity Learn · Accepted Answer

Magnetic field at the centre of a circular coil $$isB=$$μ$$04$$π × $$2$$$$π$$irwhere i is current flowing in the coil and r is radius of coil.At the centre of coil $$1$$,$$B1=$$μ$$04$$π × $$2$$$$π$$$$i1r1$$                  …$$(i)At$$ the centre of coil $$2$$,$$B2=$$μ$$04$$π × $$2$$$$π$$$$i2r2$$                  …$$(ii)But$$ $$B1$$ $$=$$ $$B2$$∴    μ$$04$$π  $$2$$$$π$$$$i1r1=$$  μ$$04$$π  $$2$$$$π$$$$i2r2$$  or  $$i1r1$$  $$=i2r2As$$ $$r1$$ $$=$$ $$2r2$$∴    $$i12r2$$  $$=i2r2$$  Or  $$i1$$  $$=$$  $$2i2$$        …$$(iii)Now$$, ratio of potential $$differencesV2V1=$$ $$i2$$ ×  $$r2i1$$ ×  $$r1$$  $$=i2$$ ×  $$r22i2$$ ×  $$2r2$$ $$=14$$∴    $$V1V2=$$ $$41$$

Two circular coils 1 and 2 are made from the same wire but the radius of the 1st coil is twice that of the 2nd coil. What is the ratio of potential difference applied across them so that the magnetic field at their centres is the same ?

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