Suppose the current in the infinite long straight circuit C of Figure 13-5 is given by, see attached, Find the induced emf that will be produced in the rectangular circuit of this same figure. What is that direction of the induced current.© BrainMass Inc. brainmass.com October 2, 2022, 4:53 am ad1c9bdddf
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Magnetic flux density at a distance x from the infinite current carrying conductor is given by:
B = μoI'/2Пx
Let us consider a strip of infinitesimally small width dx as shown in the fig..
Area of the strip = dA = bdx
Flux through the strip = dφ = BdA = [μoI'/2Пx]bdx
Flux through the rectangular circuit = φ = [μoI'b/2П]∫(1/x)dx
φ = [μoI'b/2П][loge(d+a) - loged] = [μoI'b/2П]loge[(d+a)/d]
Substituting for I' we get: φ = [μo(Ioe-λt)b/2П]loge[(d+a)/d]
φ = [μoIob/2П]loge[(d+a)/d] e-λt
By Faraday's law of electromagnetic induction, induced emf in the circuit is equal to the negative rate of change of flux linking with the circuit:
e = - dφ/dt = [μoIob/2П]loge[(d+a)/d]λ e-λt
e = [μoIoλb/2П]loge[(d+a)/d] e-λt
As the current in the infinite conductor is decreasing, by Lenz's law, the induced current in the rectangular circuit will be such as to produce a magnetic field which opposes the decrease in the magnetic field due to I' i.e. is in the same direction. By right hand rule (curl the fingers of the right hand in the direction of current flow, then the stretched out thumb gives the direction of the magnetic field lines) the direction of induced current in the circuit is clockwise.© BrainMass Inc. brainmass.com October 2, 2022, 4:53 am ad1c9bdddf>