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- A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by n from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by J=J0k, where J0 is a constant and k is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R1and current density Jonto the field of a solid cylindrical conductor with radius R2and current density J . Then use the fact that the appropriate azimuthal unit vectors can be expressed as 1=kr1and 2=kr2 to show that everywhere inside the cavity the magnetic field is given by the constant B=120J0ka , where a=r1r2 and r1=r1r1 is the position of P relative to the center of the conductor and r2=r2r2 is the position of P relative to the center of the cavity.A reasonably uniform magnetic field over a limited region of space can be produced with the Helmholtz coil, which consists of two parallel coils centered on the same axis. The coils are connected so that they carry the same current I. Each coil has N turns and radius R, which is also the distance between the coils, (a) Find tire magnetic field at any point on the z-axis shown in the accompanying figure, (b) Show that dB/dz and d2Bdz2 are both zero at z = 0. (These vanishing derivatives demonstrate that the magnetic field varies only slightly near z = 0.)The normal to tt plane of a single-turn conducting loop is directed at an angle to a spatially uniform magnetic field B. It has a fixed area and orientation relative to the magnetic fleck Show that the emf induced In the loop is given by i = (dB/dt)(Acos),where A is the area of the loop.
- Would a closed semi-circle conductor be a planar Faraday's cage?Multiple-choice question: As shown in the figure, there is an infinitely long flat conductor piece of width a and regardless of thickness, through which a uniformly distributed current I flows, and is on the right side of the conductor pieceThere is a point P coplanar with the conductor sheet at edge b, then the magnetic induction intensity B of point P isA dielectric material with μ = μo, ε = 9εo σ = 4 S/m is placed between the plates of a parallel-plate capacitor. Calculate the frequency at which the conduction and displacement currents are equal.
- An upright rectangular loop of height a and width b has a lightbulb attached to it, with some resistance R. It is immersed in a constant magnetic field with magnitude B0, pointing into the page. Now you want to turn on the lightbulb by using Faraday’s law of electromagnetic induction. Since the magnetic field is fixed and the dimensions of the loop are also fixed, the only way to change the magnetic flux in the loop is to rotate it, let’s say at some angular speed ω. Write down the value of the magnetic flux as a function of time.The current I through a long solenoid with n turns per meter and radius R is changing with time as given by dI/dt. Calculate the induced electric field as a function of distance r from the central axis of the solenoid.Consider an infinite hollow conducting cylinder of inner radius R and outer radius 3R, as shown. The non-uniform current density J is out of the page and varies with distance r fromthe center as J=J0rk (k is k hat) where J0 is a positive constant. Calculate the magnetic field at point P (r = 2R) from the centre,(magnitude and direction). Sketch the Amperian loop.
- A thin conducting ring of radius a lies in the x-y plane and carries an alternating current I = I0 cos(ωt). a. Calculate the electric dipole moment. b. Calculate the magnetic dipole moment. c. Calculate the electric field in the radiation zone. d. Calculate the total time-averaged power radiated.Can you distinguish between Gauss law in electrostatics and magnetism?Specifically, explain the reason, why divergence of magnetic field is zero?A straight conductor carrying current i=5.0 A splits intoidentical semicircular arcs as shown in Fig. 2. What is themagnetic field at the center C of the resulting circular loop?Please explain your result.