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16 good questions on electromagnetism

(See attached file for full problem description with proper symbols and diagrams)

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3.1 A long wire carries a current of 2 A along the +z axis. Calculate B in free space at (3,4,9).

3.2 Inside a long conductor of radius a, the magnetic field is H = 5 rho;/(2pi a2) a(phi) A/m. (a) Determine the current density J. (b) Determine the total current carried by the conductor.

3.3 A three-conductor transmission line is configured as shown below. Determine the magnitude and the direction of the magnetic field H at P.

3.5 A transmission line is composed of two parallel (infinitely) long conductors spaced 2-m apart in air. (a) If the magnetic field H at the midpoint between the lines has a magnitude of 100 A/m, determine the current in each conductor of the transmission line. (b) If the direction of the current in each conductor were reversed because of incorrect wiring, how would the magnitude and the direction of the magnetic field at the midpoint change?

3.7 A square loop (0.02 m x 0.02 m) has 100 turns and it lies in the x-y plane. It carries a current of 5 mA. If the loop is exposed to a magnetic flux density B = 2 x 10-3 a Wb/m2 , determine: (a) the magnetic force exerted on the loop; (b) the maximum torque experienced by the loop; (c) If the loop is free to rotate, sketch its equilibrium position.

3.8 A transmission line is composed of two parallel (infinitely) long conductors spaced 2-m apart in air. The conductors carry equal and opposite currents of 100 A each. (a) Determine the magnetic field H (magnitude and direction) at the midpoint between the lines. (b) If both conductors were carrying currents in the same direction, how would the magnitude and the direction of the magnetic field at the midpoint change?

3.10 Two circular loops are centered at the origin in the x-y plane in air. The inner loop has a radius a = 5 cm and it carries a counterclockwise current I1 = 2.5 A. The outer loop has a radius b = 8 cm and it carries a current I2. (a) If the magnetic field at the origin is measured to be zero, determine the magnitude and the direction of I2. (b) If the magnitude of the current in each loop were doubled, how would the magnitude and the direction of the magnetic field at the origin change? Explain.

3.11 A toroid has the following parameters: radius R = 0.1 m; coil radius r = 0.01 m; number of turns N = 1,000; core material = air. (a) If the toroid carries a current of 10 mA, determine the stored magnetic energy. (b) What should be the relative permeability of the core, if we want to increase the stored energy by a factor of 1,000?

3.12 Two long straight wires carry currents of 2 A each in the + a¬z direction. The wires cross the x-y plane at (0, 0, 0) and (4, 0, 0). Determine the magnetic field vector H at: (a) (2, 0, 0); (b) (8, 0, 0).

3.14 Two long straight wires carry currents of 12 A each in the + ax direction. The wires cross the y-z plane at (0, 0, -4) and (0, 0, 4). Determine the magnetic field vector H at: (a) (2, 0, 0) and (b) (0, 0, 8).

3.15 The magnetic field H is given as: H = J0 /3 a A/m for  a and H = J0 a/(3  a A/m for  a. (a) Determine the current density J everywhere. (b) Determine the total current I passing through the annular region a b, located in the x-y plane.

4.1 A square loop (0.1 m x 0.1 m) has 3 turns and it lies in the x-y plane. If the loop is exposed to a time-varying magnetic flux density B = 2 sin (120 pi t) Wb/m2 , determine the maximum value of the induced emf.

4.2 A circular metal loop (radius = 0.1 m) has 100 turns and it lies in the x-y plane. If the loop is exposed to a magnetic flux density B = 2 cos (100 pi t) Wb/m2 , determine: (a) the magnitude of the maximum emf induced in the loop; (b) the direction of current flow during 0 < t < 1/100 s.

4.3 A square loop (0.2 m x 0.2 m) lies in the x-y plane. When the loop is exposed to a magnetic flux density B = 5 x 10-3 cos (2 pi t) Wb/m2 , a sinusoidal emf with a peak value of 62.8 mV is induced in the loop. (a) Determine the frequency f. (b) If the loop had 10 turns, how would the induced emf change?

4.4 A rectangular loop (0.3 m x 0.2 m) has 10 turns and it lies in the y-z plane. It carries a counterclockwise current of 1 A. The loop is exposed to a magnetic flux density B = 5 x 10-3 Wb/m2. (a) Determine the magnitude and the direction of the torque experienced by the loop. (b) If the loop is free to rotate, sketch its equilibrium position.

4.5 A rectangular loop (0.3 m x 0.2 m) has 3 turns and it lies in the x-y plane. If the loop is exposed to a magnetic flux density B = 2 sin (1000 pi t) Wb/m2, determine: (a) the maximum value of the induced emf and (b) the direction of current flow at t = 0. Explain.
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Solution Preview

Please see the attachment.

3.1 A long wire carries a current of 2 A along the +z axis. Calculate B in free space at (3,4,9). (F-00, Exam #2)
Answer:
Magnetic field due to a current carrying conductor is given by Biot-Savart rule and the derivation for field at a point P at (perpendicular) distance r from an infinitely long current carrying wire is given in your text. As the point P is (3,4,9) r is (32 +42) = 5m. The magnitude of this field is given as
and its direction is perpendicular to r given by ampere's right hand rule.
'If the thumb of right hand is showing the direction of current then the fingers shows the direction of magnetic flux (field).'
The direction is shown in the figure.
The field at P may be given as T.
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3.2 Inside a long conductor of radius a, the magnetic field is H = 5 /(2a2) a A/m. (a) Determine the current density J. (b) Determine the total current carried by the conductor. (F-00, Exam #2)

Answer:
(I think r is the distance r from the origin, you have not given meaning of notation)
Using circuital law , where I is the current within the loop we get for a circular loop
H2r = I or H = I/2r where I is the current within the loop of radius r. gives
I = 2rH.
If the current within the loop of radius is r+dr be I+dI and the field is H +dH then
H + dH = (I+dI)/2(r+dr) gives us
I+dI =2 (r+dr)(H + dH) = 2(rH+Hdr+rdH) {drdH is very small}
Subtracting we get
dI = 2 (Hdr +rdH)
Now if the current density at distance r from the center is j, then
J = dI/2 rdr (where 2 r dr is the area of the element between r and r+dr)
Or j = (H/r + dH/dr). ....................................(1)
But H = 5 r/(2a2)a A/m or H = 5 r/(2a2) gives dH/dr = 5 /(2a2), hence
J = 5/(2a2) +5/(2a2) = 10/(2a2) constant.
Total current carried by the conductor is J.a2 = 5 A/m2.
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3.3 A three-conductor transmission line is configured as shown below. Determine the magnitude and the direction of the magnetic field H at P. (F-01, Exam #2)
Answer:

The field due two the three wires at P is along y axis and hence we can find the resultant field just by the algebraic sum.

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3.5 A transmission line is composed of two parallel (infinitely) long conductors spaced 2-m apart in air. (a) If the magnetic field H at the midpoint between the lines has a magnitude of 100 A/m, determine the current in each conductor of the transmission line. (b) If the direction of the current in each conductor were reversed because of incorrect wiring, how would the magnitude and the direction of the magnetic field at the midpoint change? (S-02, Exam #2)
Answer:
a) The current in the wires of the transmission line is in opposite direction hence using right hand rule we find that the field at midpoint are in same direction and hence

b) If the direction of the current in each wire is reversed the field will be in opposite direction.
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3.7 A square ...

Solution Summary

16 good problems of electromagnetic theory related to force, torque, magnetic field, parallel wires, current, current density, transmission line, square loop, circular loop, toroid, induced EMF, induced current are solved.

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