Two concentric coplanar circular loops made of wire with resistance per unit length $$10$$−$$4$$Ω$$/m$$, have diameters $$0.2$$ m and $$2$$ m. A time varying potential difference (4$$ $$+$$ $$2.5t) volt is applied to the larger loop. The current (in$$ $$A) in the smaller loop is $$_________________$$.

Question

Two concentric coplanar circular loops made of wire with resistance per unit length $$10$$−$$4$$Ω$$/m$$, have diameters $$0.2$$ m and $$2$$ m. A time varying potential difference (4$$ $$+$$ $$2.5t) volt is applied to the larger loop. The current (in$$ $$A) in the smaller loop is $$_________________$$.

Infinity Learn · Accepted Answer

The situation described in question is shown in figureThe magnetic induction at O, due to the current in large loop is given as where  $$B0=$$μ$$0i2R$$⇒ $$i=VR=(4+2.5t)(2$$$$π$$$$R)$$ρ⇒ $$B0=$$μ$$02R(4+2.5t)(2$$$$π$$$$R)$$ρHere we consider that the central loop is very small i.e., r << R, so the field may be considered as constant within the loop. Thus within the loop the field is $$B0$$ so the flux through smaller loop is given asϕ$$=B0$$×π$$r2=$$μ$$02R(4+2.5t)(2$$$$π$$$$R)$$ρ⋅π$$r2Induced$$ emf in the smaller loop is given $$ase=d$$ϕ$$dt=$$μ$$0r24R2$$ρ×$$(2.5)Thus$$ current i in the smaller loop is given $$asi=eR=$$μ$$0r24R2$$ρ×$$2.5$$×$$12$$$$π$$rρ⇒ $$i=2.5$$μ$$0r8$$$$π$$$$R2$$ρ$$2$$⇒$$i=2.5$$×$$4$$π×$$10$$−$$7(0.1)8$$π×$$1$$×$$10$$−$$8=1.25A$$

Two concentric coplanar circular loops made of wire with resistance per unit length 10−4Ω/m, have diameters 0.2 m and 2 m. A time varying potential difference (4 + 2.5t) volt is applied to the larger loop. The current (in A) in the smaller loop is _________________.

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