A rectangular solid conducting bar having a square cross-section of side \"a\" a
ID: 2019204 • Letter: A
Question
A rectangular solid conducting bar having a square cross-section of side "a" and length L is placed near the surface of the Earth at the top of a pait of smooth parallel superconducting rails forming a ramp angle . THe bar is made of frictionless material having a mass per unit volume (density) D and a resistivity . The rails are separated by the distance L and interconnected by a superconducting section at the bottom such that a conducting loop is created when the bar is positioned on the rails. A verical magnetic filed B having a constant direction along -j but changing magnitude exists between section and the bar remains at rest. g is the freefall rate of acceleration and t is the time such that |B| = B0 when t = 0...
Determine an experssion for B in terms of t, S, , g, D and
Explanation / Answer
The bar will keep accelerating until the force from the current generated cancels the force of gravity. The force of gravity on the bar pointing down the rails is F = IBL The component of magnetic field normal to the block B = Bcos F = IBLcos The component of gravitational force acting on the bar mgsin = IB0Lcos ==> B0 = mgtan / IL density of the rod D = m / aL ==> m = mass of the rod = DaL I = DaLgtan / B0L ==> Dagtan / B0 ..........(1) The induced emf produced along the length of the bar = change in magnetic flux = d /dt = A(dB /dt) Induced emf = IR , A = change in magnetic flux threough an area = sL Here I = induced current , R = resistance of the bar = L / a I (L / a) = A(dB /dt) (dB /dt) = I(L / asL) dB = I(L / asL) dt intergrating on both sides we get B(t) = I(L / asL)t From equation (1) I = Dagtan / B0 Hnce B (t) = (Dagtan / B0 ) (L / asL)t = = (Dagtan / B0 ) ( / as)t B = Bcos F = IBLcos The component of gravitational force acting on the bar mgsin = IB0Lcos ==> B0 = mgtan / IL density of the rod D = m / aL ==> m = mass of the rod = DaL I = DaLgtan / B0L ==> Dagtan / B0 ..........(1) The induced emf produced along the length of the bar = change in magnetic flux = d /dt = A(dB /dt) Induced emf = IR , A = change in magnetic flux threough an area = sL Here I = induced current , R = resistance of the bar = L / a I (L / a) = A(dB /dt) (dB /dt) = I(L / asL) dB = I(L / asL) dt intergrating on both sides we get B(t) = I(L / asL)t From equation (1) I = Dagtan / B0 Hnce B (t) = (Dagtan / B0 ) (L / asL)t = = (Dagtan / B0 ) ( / as)tRelated Questions
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