VDU_Physics_3_011

# ## Solved University-Physics Problems

Physics 3:  Introduction to Electricity & Magnetism

## <<<  Induced Magnetic Field.    Ampere's Law  >>>

Javier Montenegro Joo

jmj@VirtualDynamicsSoft.com

Return to Electricity and Magnetism Index page

## <<<  Induced Magnetic Field.    Ampere's Law  >>>

(1) A long straight wire of radius R transports an electric current. Calculate the radial distance where the induced magnetic field due to this current, reduces to a half of its maximum value. (a) Out of the wire  (b) inside the wire.

Solution

The maximum induced magnetic field due to an electric current in a wire, is at the surface of the wire.  This is valid for both magnetic fields, exterior and interior.  In this problem the surface of the wire is at  r = R.

The magnetic field induced by a current in a wire is: (2) The propulsion system of an interstellar-travel vessel includes a long, straight, solid, cylinder of radius R.  The cylinder leads lengthwise an electrical current I whose density is  J = A r,  A = Const.  Determine: (a) the constant A  (b) the magnetic field in r ≤ R  (c) the magnetic field in r ≥ R.

Solution.- Oscillations in a rectangular membrane:

The EduVirtualLab about oscillations in a rectangular membrane shows how to make the simulation module This simulation module is included in the Physics Virtual Lab (PVL):

(3)  A cylindrical stream of ions (charged particles) has a transversal section of radius R and it transports an electrical current   Io whose density    J = Ao r².    Determine: (a) the constant Ao (b) The current between  r = R/2   and  r = R (c)  Current between   r = 0  and   r = R/2  (d) Verify that the addition of the currents obtained in (b) and (c) equal the total current Io.

Solution.-

Note that when adding the results of parts (b) and (c) the total electric current  Io must be obtained. (4) A cylindrical stream of charged particles has radius R and a density of electric current that varies with the radius according to J = Ao r, where Ao is a constant and r is the radial distance measured from the axis of the stream. Determine the magnetic field induced in:  (a) r > R         (b)  r < R.

Solution.-

The magnetic field induced by the stream of charged particles in the region exterior to the stream depends on the total current in the cylinder, while the field induced in its interior region depends on a fraction of the total current. (5) The sketch shows two parallel vertical cables on the plane XY. The cables are separated by a distance R and, each one transports an electrical current Io in opposite direction.  Determine the magnetic fields resulting at points A, B and C. Solución.-

Since the XY-plane is on the plane of the sketch, the z-axis is coming out the XY-plane. The sketch shows the transversal cut of four parallel wires separated a distance 2L from each other. The wires identified as 1, 2, 3 and 4, transport each an electric current whose directions and magnitudes are shown. Determine the magnetic field induced by each current at the center of the square, find also the total magnetic field at the same point. Solution

The circular line of magnetic field that surrounds each wire is tangent to the center of the square, hence the magnetic field due to each current is also tangent at that point. The direction of each magnetic field at the center of the square is deduced from the direction of each current. PVL: Physics Virtual Lab. 