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Physics 1551 (2023) Assignment 8 Due: Thursday, March 30 Hand in to drop slots on second floor of Dunn. Answer in the spaces provided. Your name (print: Last, First) Your Mark (out of 16) 1. Applying EM Induction to Medicine. A stent is a cylindrical tube, often made of metal mesh, that's inserted into a blood vessel to overcome a constriction. It's sometimes necessary to heat the stent after insertion to prevent cell growth that could cause constriction to recur. One method is to place the patient in a changing magnetic field, so that induced currents heat the stent. Consider a stainless-steel stent 12 mm long by 3.5 mm in diameter, with total resistance 41 m Ω . Treating the stent as a single wire loop in the optimum orientation, find the rate of change of magnetic field needed for a heating power of 210 mW [4] Hint: Remember that the power dissipated by a resistor can be given as 2. “Wireless” charging An electric toothbrush uses induction to charge its battery. We can model this as a setting up one solenoid inside another. If we choose two solenoids of the same length but different diameters and number of loops: (a) Find an expression for the magnetic field inside the inner solenoid in terms of the applied current I 1 , the number of coils N 1 , the diameter d 1 , and the length of the solenoids l . [1] (b) Find an expression for the flux experienced by the outer coil which has N 2 coils, diameter d 2 , and length l . (This coil would be located in the base of the toothbrush). [2] (c) Given that L1=6.5μ H , d 1=2.5 mm , d 2=8.0 mm , and l=4.4 mm , find the total number of coils needed for the second solenoid if the driving current I 1=(0.060 A)cos(ω t ) is expected to produce a maximum voltage of 3.6 V . Note that the frequency of the driving current is f =60 Hz . [3] In reality the number of coils is significantly smaller, as we can use an iron core to increase the inductance of the coils. 3. Eddy currents. A conducting disk with radius a, thickness h, and resistivity ρ is inside a solenoid of circular cross section. The disk axis coincides with the solenoid axis. The magnetic field in the solenoid is given by B=bt , where b is a constant and t is the time. Find expressions for: (a) The current density J (current/area) in the disk as a function of the distance r from the disk center [3] (b) The rate of power dissipation in the entire disk. [3] Hints: Use Ohm’s Law to connect the induced EMF to the current density, where V =IR , J ≡I / A , and the resistivity along a length L is defined as ρ≡RA / L . Consider the disk to be made up of infinitesimal conducting loops of width dr . The situation is similar to the aluminium plate that is dragged through the magnetic field demonstrated in class, but easier to calculate.

Physics 1551 (2023) Assignment 8 Due: Thursday, March 30 Hand in to drop slots on second floor of Dunn. Answer in the spaces provided. Your name (print: Last, First) Your Mark (out of 16) 1. Applying EM Induction to Medicine. A stent is a cylindrical tube, often made of metal mesh, that's inserted into a blood vessel to overcome a constriction. It's sometimes necessary to heat the stent after insertion to prevent cell growth that could cause constriction to recur. One method is to place the patient in a changing magnetic field, so that induced currents heat the stent. Consider a stainless-steel stent 12 mm long by 3.5 mm in diameter, with total resistance 41 m Ω . Treating the stent as a single wire loop in the optimum orientation, find the rate of change of magnetic field needed for a heating power of 210 mW [4] Hint: Remember that the power dissipated by a resistor can be given as 2. “Wireless” charging An electric toothbrush uses induction to charge its battery. We can model this as a setting up one solenoid inside another. If we choose two solenoids of the same length but different diameters and number of loops: (a) Find an expression for the magnetic field inside the inner solenoid in terms of the applied current I 1 , the number of coils N 1 , the diameter d 1 , and the length of the solenoids l . [1] (b) Find an expression for the flux experienced by the outer coil which has N 2 coils, diameter d 2 , and length l . (This coil would be located in the base of the toothbrush). [2] (c) Given that L1=6.5μ H , d 1=2.5 mm , d 2=8.0 mm , and l=4.4 mm , find the total number of coils needed for the second solenoid if the driving current I 1=(0.060 A)cos(ω t ) is expected to produce a maximum voltage of 3.6 V . Note that the frequency of the driving current is f =60 Hz . [3] In reality the number of coils is significantly smaller, as we can use an iron core to increase the inductance of the coils. 3. Eddy currents. A conducting disk with radius a, thickness h, and resistivity ρ is inside a solenoid of circular cross section. The disk axis coincides with the solenoid axis. The magnetic field in the solenoid is given by B=bt , where b is a constant and t is the time. Find expressions for: (a) The current density J (current/area) in the disk as a function of the distance r from the disk center [3] (b) The rate of power dissipation in the entire disk. [3] Hints: Use Ohm’s Law to connect the induced EMF to the current density, where V =IR , J ≡I / A , and the resistivity along a length L is defined as ρ≡RA / L . Consider the disk to be made up of infinitesimal conducting loops of width dr . The situation is similar to the aluminium plate that is dragged through the magnetic field demonstrated in class, but easier to calculate.

Answered 1 days AfterMar 29, 2023

P = I2R

210 X 10-3 = I2(41

I = 2.26A

And the induced EMF is calculated as = ΔBA/Δt

Under optimum orientation, area is equal to the circle of stent

Area = πr2 = π (1.75 X 10-3)2 = 0.96 X 10-5 m2

EMF is calculated using IR as

IR = ΔBA/Δt

Now, to the rate of change of the magnetic field ΔB/Δt is calculated as

ΔB/Δt = IR/A = (2.26) (41 X 10-3)/0.96 X 10-5

ΔB/Δt = 9.65 X 103 T/s

Now, E = dΦ/dt = d(BA)/dt = AdB/dt

also, P = E^2/R

so, P = (AdB/dt)2 /R

so, dB/dt = √(PR/A2)

= √(210*10-3*41*10-3/(π*(1.75)2)2)

= 0.0966 Tesla /sec

Ans 2 (a) The magnetic field inside the solenoid is calculated as:

B = μ₀ N I / L

here μ₀ = permeability of free space,...

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