Problem Set 4 Magnetic Force An electron with

Problem Set 4

Magnetic Force • An electron with a speed of 8 × 106 m/s is projected along the positive x-direction into a medium containing a uniform magnetic flux density B = (x 4 − z 3) T. Given that e = 1. 6 × 10− 19 C and the mass of an electron is me = 9. 1× 10− 31 kg, determine the initial acceleration vector of the electron (at the moment it is projected into the medium). The acceleration vector of a free particle is the net force vector divided by the particle mass. Neglecting gravity;

Magnetic Force • The rectangular loop shown in Fig. consists of 20 closely wrapped turns and is hinged along the z-axis. The plane of the loop makes an angle of 30◦ with the y-axis, and the current in the windings is 0. 5 A. What is the magnitude of the torque exerted on the loop in the presence of a uniform field B = y 2. 4 T? When viewed from above, is the expected direction of rotation clockwise or counterclockwise?

Magnetic Force • T = m×B where m is the magnetic moment given by • m= n. NIA • n=−xcos 30◦+ysin 30◦ • m = n 0. 8 • T = n 0. 8 × y 2. 4 • T=0. 8(−xcos 30◦+ysin 30◦) × y 2. 4 • T=−z 1. 66 (Nm)

Magnetic Force • In a cylindrical coordinate system, a 2 m-long straight wire carrying a current of 5 A in the positive z-direction is located at r = 4 cm, φ = π/2, and − 1 m ≤z≤ 1 m (a) If B = r 0. 2 cosφ (T), what is the magnetic force acting on the wire? (b) How much work is required to rotate the wire once about the z-axis in the negative φ-direction (while maintaining r = 4 cm)? (c) At what angle φ is the force a maximum?

Magnetic Force

The Biot Savart Law • An infinitely long wire carrying a 25 -A current in the positive xdirection is placed along the x-axis in the vicinity of a 20 -turn circular loop located in the x–y plane as shown in Fig. If the magnetic field at the center of the loop is zero, what is the direction and magnitude of the current flowing in the loop?

The Biot- Savart Law • Magnetic field of the wire • Since the net field is zero at the center of the loop, I 2 must be clockwise

The Biot- Savart Law • Two parallel, circular loops carrying a current of 40 A each are arranged as shown in Fig. The first loop is situated in the x-y plane with its center at the origin and the second loop's center is at z = 2 m. If the two loops have the same radius a = 3 m, determine the magnetic field at: v (a) z = 0 (b) z =1 m (c) z =2 m

The Biot- Savart Law

The Biot- Savart Law • The current direction is along –φ • For the second loop, which is at a height of 2 m z should be replaced with (z-2)

The Biot- Savart Law • The total field is • A) at z=0 • B) at z=1 • C) at z=2 H should be the same as at z=0

Ampere’s Law • A long cylindrical conductor whose axis is coincident with the z-axis has a radius a and carries a current characterized by a current density where J 0 is a constant and r is the radial distance from the cylinder’s axis. Obtain an expression for the magnetic field H

Ampere’s Law

Ampere’s Law • In a certain conducting region, the magnetic field is given in cylindrical coordinates by • Find the current density J

Magnetic Boundary Conditions • The x–y plane separates two magnetic media with magnetic permeabilities μ 1 and μ 2, as shown in Fig. If there is no surface current at the interface and the magnetic field in medium 1 is Find • A) H 2 • B) θ 1 and θ 2

Magnetic Boundary Conditions

Magnetic Boundary Conditions • Given that a current sheet with surface current density Js = x 8 (A/m) exists at y=0. the interface between two magnetic media, and H 1= z 11 (A/m) in medium 1 (y>0), determine H 2 in medium 2. • H 1 is tangential to the boundary, and therefore and so is H 2

Magnetic Boundary Conditions since μ 1 H 1 y = μ 2 H 2 y H 1 does not have a y-component H 2 does not have an x component

Magnetic Boundary Conditions • Show that if no surface current density exists at the parallel interfaces shown in Fig , the relationship between θ 4 and θ 1 is independent of μ 2.

Magnetic Boundary Conditions

Magnetic Energy • In terms of the d-c current I, how much magnetic energy is stored in the insulating medium of a 3 -m-long, air-filled section of a coaxial transmission line, given that the radius of the inner conductor is 5 cm and the inner radius of the outer conductor is 10 cm? the inductance per unit length of an air-filled coaxial cable is given by

Magnetic Energy • Magnetic energy is given as

• A circular loop of radius a carrying current I 1 is located in the x–y plane as shown in the figure. In addition, an infinitely long wire carrying current I 2 in a direction parallel with the z-axis is located at y • Determine H at (0, 0, h)