Magnetic Sources and Induction Homework Set

Size: px

Start display at page:

Download "Magnetic Sources and Induction Homework Set"

Kristin Marshall
7 years ago
Views:

1 Problem 1. Use the Biot-Savart law to calculate the magnetic field B at point C, the common center of semi-circular arcs AD and HJ respectively in the figure below. The two arcs, of radii R 1 and R 2, respectively, form part of the circuit ADJHA carrying current i. R 2 i i R 1 C A H J D In the figure above, use two colors to show the direction of the wire segment and the direction from the wire segment to point C. From this drawing make arguments based on the properties of the vector product about which wire segments (the two semi-circular segments and the two linear segments) which will contribute to the magnetic field.

2 In the space below, write the integrand that will be used. For example, if the linear segments will contribute then dl=dx. Integrate the contributions from the appropriate wire segments in the space below. Use the principle of superposition to find the total magnetic field at point C.

3 Problem 2. The figure shows five long parallel wires in the xy plane. Each wire carries a current i=3.0 A in the positive x direction. The separation between adjacent wires is d= 8.00 cm. In unit-vector notation, what is the magnetic force per meter exerted on each of these five wires by the other wires? z d d d d y Label the wires wire 1 through 5 from left to right. In the space below, draw the field lines from the other wires which are passing through wire 1. Be sure to use the right-hand rule Repeat these drawings for the other four wires in the space below.

4 For wire 1, calculate the magnitude of the magnetic emanating from the other wires in the space below. Label these as B 12, B 13, etc. Using the drawings of the field lines and their magnitudes, calculate the magnetic field at wire 1 from the other wires using the principle of superposition. Use the force law to calculate the force on wire 1.

5 For wire 2, calculate the magnitude of the magnetic emanating from the other wires in the space below. Label these as B 22, B 23, etc. Using the drawings of the field lines and their magnitudes, calculate the magnetic field at wire 2 from the other wires using the principle of superposition. Use the force law to calculate the force on wire 2.

6 For wire 3, calculate the magnitude of the magnetic emanating from the other wires in the space below. Label these as B 31, B 32, etc. Using the drawings of the field lines and their magnitudes, calculate the magnetic field at wire 3 from the other wires using the principle of superposition. Use the force law to calculate the force on wire 3.

7 For wire 4, use symmetry arguments to calculate the force on this wire. For wire 5, use symmetry arguments to calculate the force on this wire.

8 Problem 3: In the figure, the long straight wire carries a current of 30 A and the wire loop carries a current of 20 A. Calculate the resultant force acting on the loop. Assume that a=1.0 cm, b= 8.0 cm and L=30 cm. 30A a 20A b 20A L Below, use the RH-rule and re-draw the diagram with the direction of the magnetic fields shown on each wire segment.

9 Below, use the RH-rule and create a free body diagram for each wire segment. Use a positive value for forces which are up and to the right on the page. Calculate the magnitude of the forces which are not eliminated by the free body diagram.

10 Perform algebraic notation to solve the problem.

11 Problem 4. The figure below shows a cross-section of a hollow cylindrical conductor of radii a and b, carrying a uniformly distributed current i. a) Show that B(r) for the range b< r <a is given by a r b Calculate the cross-sectional area of the hollow cylinder. Find the current density for the hollow cylinder. (Hint: J=i/A) Calculate the cross-sectional area of hollow cylinder whose inner radius is b and whose outer radius is r. Find the current enclosed in a hollow cylinder whose inner radius is b and whose outer radius is r using its cross-sectional area and the constant current density derived above. (Hint: i=ja)

12 With the above current enclosed, use Ampere s Law to calculate the magnetic field B(r). b) Show for r=a that this expression reduces to a magnetic field of a long straight wire Perform the substitution and calculate the field. Derive the magnetic field of a long straight wire with current i using Ampere s law. Compare the above results.

13 c) Show for r=b that this expression give zero magnetic field. Perform the substitution and calculate the field.

14 Problem 5. A solenoid 1.30 meters long and 2.60 cm in diameter carries a current of 18 A. The magnetic field inside the solenoid is 23.0 mt. Find the length of the wire forming the solenoid. Convert relevant quantities to SI units. Use the expression of the magnetic field for a solenoid to calculate the total number of turns, N. Calculate the total length of wire by calculating N circumferences.

15 Problem 6. A long solenoid with a radius of 25 mm has 100 turns/cm. A single wire of wire of radius 5.0 cm is placed around the solenoid, the central axes of the loop and solenoid are coinciding. In 10 ms, the current in the solenoid is reduced from 1.0 A to 0.5 A at a uniform rate. What emf appears in the loop? Convert relevant quantities to SI units. Write the expression of the magnetic field of a solenoid. Calculate the rate of change in the current in A/s. With the rate of change current, calculate the rate of change of the magnetic field in the solenoid. Using the properties of solenoid calculate the area wherein there is a magnetic field.

16 Write an expression for the emf in terms of a constant area and a changing magnetic field. Use the quantities calculated to find the emf.

17 Problem 7. A square loop with 2.0 m sides is perpendicular to a uniform magnetic field with half of the area of the loop in the field as shown below. The loop contains a 20.0 V battery with a negligible internal resistance. If the magnitude of the field varies with time according to B= t with B in teslas and t in seconds. a) What is the total emf in the circuit? Calculate the area of the loop within the field. Calculate the magnetic flux in the loop. (Hint: Φ=BA) Find the derivative of the flux with respect to time. Find the emf generated by the magnetic field.

18 b) What is the direction of the current through the battery? In the space below, give the reasoning for the direction of the emf. (Hint: a negative result is a clockwise rotation). Based on the direction of the emf, find the total emf in the loop.

19 Problem 8. A stiff wire bent into a semicircle of radius a is rotated with a frequency f in a uniform magnetic field as suggested below. What are the a) frequency and b) amplitude of the varying emf induced in the loop. Calculate the area of the loop. Write a general expression for a sinusoidal time varying function in terms of a cosine, the linear frequency, and a phase angle φ 0. Write an express for the magnetic flux in terms of the magnetic field, the area, and a sinusoidal function. Find the time derivative of magnetic flux and use it to determine emf.

20 From this expression determine the frequency and the amplitude below. (Hint: if f(t) =A*cos(2πft +φ 0 ), then the amplitude of this function is A and its frequency is f.)

21 Problem 9. The vertical component of Earth s magnetic field in Tucson Arizona is 43 μt. Assume that this is the average value of all of Arizona which has an area of 295,000 km 2 and calculate the net magnetic flux through the rest of Earth s surface (the entire surface excluding Arizona). Is the flux outward or inward? Convert the relevant quantities to SI units. Calculate the magnetic flux through Arizona (Hint: Φ = B*A.) In the space below, draw the field lines emanating from a simple bar magnet. Draw a spherical (in 2-d circular) Gaussian surface around it in another color. The net number of field lines should be zero. Give an explanation of why. If the net number of field lines is zero, there is a simple relationship between the number of field lines going into a surface and the number of field lines going out of a surface. Write that expression below. If the field lines going through Arizona are all inward, by the reasoning in the last statement, calculate the net magnetic flux for the rest of Earth and its direction.

22 Problem 10. A magnet in the form of a cylindrical rod has a length of 5.0 cm and a diameter of 1.0 cm. It has a uniform magnetization of 5.3 x 10 3 A/m. What is its magnetic dipole moment? Convert the relevant quantities to SI units. Write an expression for the magnetization of a material in terms of the volume V and the magnetic dipole moment per volume μ. Calculate the volume of the cylinder. Perform algebraic manipulation and solve.

23 Problem 11. The magnitude of the electric field between two circular plates shown in the figure 5 E = t where E in volts per meter and t in seconds. At t=0, the field below is ( ) ( ) is upwards as shown. The plate area is 4 x 10-2 m 2. For t 0, a) what is the magnitude and direction of the displacement current between the plates? Calculate the electric flux. (Hint: Φ=E*A) Find the derivative of the electric flux with respect to time, t. Write an expression for the displacement current in terms of the electric flux.

24 b) is the direction of the magnetic field clockwise or counterclockwise around the plates? Assume that a positive direction is upwards. Determine whether the displacement current is positive or negative. Place your thumb of your right hand in the direction of the displacement current. Write your answer along with a drawing in the space below.

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name

Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name Name If you think that no correct answer is provided, give your answer, state your reasoning briefly; append additional sheet of paper if necessary. 1. A particle (q = 5.0 nc, m = 3.0 µg) moves in a region