Question

Electromagnetic field

In the Lorenz gauge, the solution for the magnetic vector potential due to a current density J is given by

\vec{A}(\vec{r}, t)=\frac{\mu_{0}}{4 \pi} \int \frac{\left[\vec{J}\left(\vec{r}^{\prime}, t\right)\right]}{\left|\vec{r}-\vec{r}^{\prime}\right|} \mathrm{d}^{3} \vec{r}^{\prime}

Re-express the square-bracket notation in a different mathematical form and give a physical interpretation of the expression.(2)

A thin wire, formed into a circular loop with radius a, is placed horizontally at the origin of a right-handed coordinate system, such that the symmetry axis of the loop coincides with the z-axis, as shown in the figure. The loop carries a constant anti-clockwise current I. By considering (without any loss of generality) an observation point with spherical coordinates (r, theta) in the x-z plane (i.e. with theta=0), show that the magnetic vector potential is given by

\vec{A}(\vec{r})=\frac{\mu_{0}}{4 \pi} I a \int_{-\pi}^{\pi} \frac{\hat{\phi}^{\prime}}{\sqrt{r^{2}+a^{2}-2 r a \sin \theta \cos \phi^{\prime}}} \mathrm{d} \phi^{\prime}

where o' is the unit vector pointing in the direction of the current flow at a point in the loop with azimuth al angular coordinate o'. (You may find it helpful to

describe the transverse dimensions of the thin wire using Dirac delta functions.)

Finally, by using the approximation (1 – e)-1/2 × 1 + €/2 with e « 1, and noting that a symmetrical integral over an odd function vanishes, show that far from the loop, with r » a, the vector potential can be expressed as

\vec{A}(\vec{r})=\frac{\mu_{0}}{4 \pi} \frac{\vec{m} \times \vec{r}}{r^{3}}

where the loop magnetic dipole m is a vector pointing along the positive z-axis with magnitude I × (area of loop).(5)

Verified

### Question 44810

Electromagnetic field

5. Suppose we have a single conductor, as would be used for a wave guide, in the shape below. Would it be possible to use this as a transmission line under low frequency, such that the device size is much shorter than the wavelength? Why or why not? Hint: what are the physical requirements for a resistor,capacitor, and inductor in a TL?

Read More

### Question 44809

Electromagnetic field

4. Suppose we have a pair of parallel plates that we wish to use as a transmission line. The dielectric medium between the plates is air: €0, Ho. l is the length of the line, w is the width of the plates, and d isthe separation between the plates.
(a) Find an expression for C'.
(b) Find an expression for L'.
(c) Plot how the signal velocity v changes as a function of d.

Read More

### Question 44808

Electromagnetic field

3. Assume that we have a lossless waveguide of width a, height b surrounding a dielectric medium with€, Ho. We want to discuss an input signal with frequency w, sufficiently large for a TE11 operating mode.
(a) Find the group velocity of the resulting wave.
(b) Find the phase velocity of the resulting wave.(
(c) What conditions for a, b would allow the group and phase velocities to converge? What physical meaning does this limit have? Hint: focus on the waveguide's influence on the medium.

Read More

### Question 44807

Electromagnetic field

2. Assume that we have a transmission line with characteristic impedance zo, with an insulator between the plates withe, Ho. We continuously input a waveform with an amplitude Vo and frequency w such that the TL length I contains exactly one wavelength.Assume load impedance z, is in open condition. Assume the generator is in matched condition, i.e. the input impedance is equal to zo.
(a) Plot the forward and reverse voltage wave forms at the given timestamps on the left axes below.To is the transit time for the length of the TL. Make sure to distinguish between forward and reverse lines.
(b) Do the same as in part (a), but for the forward and reverse current wave forms on the right axes.
(c) What is To in terms of the given TL parameters?

Read More

### Question 44806

Electromagnetic field

1. Consider a plane wave propagating along the z-axis towards the boundary between two insulators, as shown below.
(a) What is I if the incident wave comes from the left?
(c) Compare the magnitudes of I in both cases. Should they be the same? Why or why not?

Read More

### Question 44593

Electromagnetic field

What are the advantages and disadvantages of AC and DC motors?

Read More

### Question 44592

Electromagnetic field

Match the following according to the type of motor load:
a. Constant power loads b. Variable torque load c. Constant torque load

Read More

### Question 44591

Electromagnetic field

Chose the best motor type which suits each of the following applications:

Read More

### Question 44590

Electromagnetic field

Explain in brief the principle of AC synchronous and induction motors.

Read More

### Question 44589

Electromagnetic field

Justify your answer for the following questions:
DC motors use permanent magnets to construct a static magnetic field.
If a motor consumes 184 kW of power but only produces 231 HP at the shaft, itscurrent operating efficiency isanswer to the nearest one.% (1 Horsepower = 745.7 Watts). Round your
c) Induction motors use commutators to transmit current to the rotor, thus producing a rotor magnetic field.
A live reading of a motor indicates that it is consuming 16.4 kVA with a Power Factor of 0.76. You also know the current operating efficiency of the motor is 89%. What is the instantaneous power output of the motor? (1 Horsepower = 745.7 Watts).

Read More

### Submit query

Getting answers to your urgent problems is simple. Submit your query in the given box and get answers Instantly.

### Submit a new Query

Success

Assignment is successfully created