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There in fairly long solenoid whose length is l and radial of its cross section is b. This solenoid is connected to a battery of emf V and a, resistor of resistance R in series with it along with a key which is initially open. A thin wire ring of radius a and resistance r is kept inside the solenoid in such a manner that axis of solenoid coincides with axis of ring. Assume thay there in no self-inductance at the ring. Find the value of radical force per unit length of the ring as a function of time if key is closed at t=0. What would be the maximum value of this radical force per unit length? |
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Answer» Solution :When current GROWS in solenoid, then FLUX linked with the ring also changes which induces emf in the ring and it causes flow of current in it. This ring is placed in magnetic feild of solenoid which is acting PERPENDICULAR to the plane of ring. When current flows in the ring, then it experiences radical FORCE everywhere on its length. We can write self-inductance L of the solenoid as follows: `L=mu_(0) n^(2)b^(2)l` ...(i) Current as a function of time in the solenoid can be written as follows: `i=(V)/(R)(1-E^((-tR)/(L)))` Magnetic field inside the solenoid can be written as `B=mu_(0)ni` `implies B=(mu_(0) nV)/(R)(1-e^((-tR)/(L)))` ...(ii) Magnetic flux linked with the ring can be written as follows: `phi=B(pia^(2))=(mu_(0)nVpia^(2))/(R)(1-e^((-tR)/(L)))` emf induced can be be written as follows: `epsilon=|(dphi)/(dt)|=(mu_(0)nVpia^(2))/(R)((R)/(L)e^((-tR)/(L)))` `epsilon=(mu_(0)nVpia^(2))/(L)e^((-tR)/(L))` We know that r is resistance of ring and hence current flowing in the ring can be written as follows: `I=(epsilon)/(r)=(mu_(0)nVpia^(2))/(rL)e^((-tR)/(L))` ...(iii) If we select a segment of length dl on the circumference of ring then radial force acting on this segment can be written as follows: dF = IdlB Hence radial force acting per unit length of the ring can be written as `(dF)/(dt)=IB` `implies (dF)/(dt)=((mu_(0)nVpia^(2))/(rL)e^((-tR)/(L)))[(mu_(0)nV)/(R)(1-e^((-tR)/(L)))]` We can also substitute value of self-inductance L from equation (i) to get the following: `implies (dF)/(dl)=((mu_(0)nVpia^(2))/(rmu_(0)pin^(2)b^(2)l)e^((-tR)/(L)))[(mu_(0)nV)/(R)(1-e^((-tR)/(L)))]` `implies (dF)/(dl)=(mu_(0) V^(2)a^(2))/(rb^(2)lR)[e^((-tR)/(L))(1-e^((-tR)/(L)))]` Above expression represents radial force acting on the ring per unit length as a function of time. To calculate maximum value of this force, we need to differentiate it and put it equal to zero. After calculation we get that above force per unit length is maximum when `e^(-tR // L) = 1 // 2`. Substituting this above relation we get maximum value of this force per unit length as follows: `implies ((dF)/(dl))_("maximum")=(mu_(0)V^(2)a^(2))/(4rb^(2)lR)` |
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