A long, straight, solid cylinder, oriented with its axis in the z−direction, carries a current whose current density is J⃗ . The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship J⃗ =2I0πa2[1−(ra)2]k^forr≤a=0forr≥a where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having units of amperes.
A)Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≥a.
B)Obtain an expression for the current I contained in a circular cross section of radius r≤a and centered at the cylinder axis.
C)Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≤a.
A long, straight, solid cylinder, oriented with its axis in the z−direction, carries a current whose...
2. (30 points) A very long, straight, solid copper cylinder of radius R (>2R) is oriented with its axis along e z-direction. The cylinder carries a current whose current density is j(r), where r is the radial distance from the cylinder axis. The current density, although symmetric about the cylinder axis, is not constant but varies with r according to 31o a) (10) Obtain an expression for the current /(in terms of Jo, r and R) flowing in a circular...
2. A modified coaxial cable consists of a solid cylinder (radius 'a') with a uniform current density and a concentric cylindrical conducting thin shell (radius 'b'). The outer and inner current have an equal magnitude, but are opposite in direction. Io (along outside) (along the axis) (off-axis view) In terms of radial distance 'r', and the relevant parameters in the diagram above, A) Derive an expression for the magnetic field inside the solid cylinder (r <a) B) Derive an expression...
Problem 4 (20 points): A long, solid conducting cylinder of radius R has a current density within it described by: (r)-C( ) for r< R where C is a constant to be determined. The total current running through the whole cylinder is I. a) Calculate an expression for the constant C, given that the total current is I. (Hint: the current density is not uniform.) b) Why can Ampere's law be used here, and what Amperian loop is appropriate? c)...
Cross-sectional View (current into page) A section of a long conducting cylinder with inner radius a and outer radius b carries a current lo that has a uniform current density, as shown in the figure above. (a) Using Ampère's law, derive an expression for the magnitude of the magnetic field in the following regions as a function of the distance r from the central axis. t. r<a ii. a<r<b (b) On the cross-sectional view in the diagram above, indicate the...
An infinitely long, straight, cylindrical wire of radius R carries a uniform current density J. Using symmetry and Ampere's law, find the magnitude and direction of the magnetic field at a point inside the wire. For the purposes of this problem, use a cylindrical coordinate system with the current in the +z-direction, as shown coming out of the screen in the top illustration. The radial r-coordinate of each point is the distance to the central axis of the wire, and...
Problem 4, 30 marks The infinitely long conducting cylinder of radius R carries the volume current density directed along its axis whose absolute value is a cubic function of the distance from the center of the cylinder r, j(r)-br3, where b is a known constant. a. Find the magnitude and direction of the magnetic field B forr>R. b. Find the magnitude and direction of the magnetic field B for r<R. c. Imagine that the conductor has magnetic permeability H (5...
5.22 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 J żJo/r, where Jo is a constant and r is the radial distance from the cylinder's axis. Obtain an expression for the magnetic field H for (a) 0<r Sa (b) r > a
The current density inside a long, solid, cylindrical wire of radius a = 4.0 mm is in the direction of the central axis and its magnitude varies linearly with radial distance r from the axis according to J = J0r/a, where J0 = 280 A/m2. Find the magnitude of the magnetic field at a distance (a) r=0, (b) r = 2.7 mm and (c) r=4.0 mm from the center. Chapter 29, Problem 047 The current density inside a long, solid,...
An infinitely long cylinder with axis aloong the z-direction and radius R has a hole of radius a bored parallel to and centered a distance b from the cylinder axis (a+b<R). The charge density is uniform and total charge/length is placed on the cylinder. Find the magnitude and direction of the electric field in the hole.
An infinite solid cylinder conductor of radius a = 3cm centered on the z-axis carries a current I1 = 1A. The current is evenly distributed along the cross section and is directed out of the screen (positive z-axis direction). An infinite coaxial conductive surface of radius b = 8 cm carries a current I2 = 4A, towards the inside of the screen (negative direction z). What is the magnitude of the magnetic field B inside the inner cylinder at a...