Problem 2 Show that in a current-free volume of space, a static magnetic field can never have a local maximum by considering V(B.B) da, where S is the surface of a small volume V containing a point P in space.
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- Consider an infinite hollow conducting cylinder of inner radius R and outer radius 3R, as shown in the image. The non-uniform current density J is out of the page and varies with distance r from the center as J = J0 r (ˆk) 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.Consider the cross section of a current carrying wire as shown below. The current is carried by the 2 gray sections (). The current, , is in the direction out of the page, half of it flowing in the inner wire (radius 0a) and half in the outer shell (between circles b and c). Assuming that the 3 sections have the same thickness, i.e. Oa = ab = bc = 1cm, calculate the magnetic field for 0 < r < a, a < r < b, b < r < c, and r > c.The arrangement illustrated in the figure below is composed of six finite straight wires of length l. The electric current flowing in such an arrangement is i. Using the Biot-Savart law, calculate: The magnitude of the magnetic field at point P due to the wire located along segment ab.
- Consider an infinite straight wire of radius R with current I.(a) Find A outside the wire.(b) Compute Del.A and Del x A outside the wire.(c) Find A inside the wire.Consider the circuit in the figure (b). The curved segments are arcs of circles of angle 35o and radii a = 12 cm and b = 6 cm. The straight segments are along radii. Find the magnetic field B at point P ( the center of curvature), assuming a current of 1 A clockwise in the circuit. (Positive out of page and negative into page) ONLY THE CURVED SEGMENTS CONTRIBUtEConsider 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.
- Problem 5.18 Show that the magnetic field of an infinite solenoid runs parallel to the axis, regardless of the cross-sectional shape of the coil, as long as that shape is constant along the length of the solenoid. What is the magnitude of the field, inside and outside of such a coil? Show that the toroid field (Eq. 5.60) reduces to the solenoid field, when the radius of the donut is so large that a segment can be considered essentially straight.Consider two infinitely long and parallel wires separated by distance d and carrying currents I₁ = -12. (a) Find the magnitude and direction of the vector potential A(r1,72) at a point P where r₁ and r2 represent the distances to P from wire 1 and wire 2 respectively. (b) What is the magnitude of A for r₁ = r₂? (c) What is the value of the magnetic field B for r₁ = r₂? (d) Given that B = V x A, how can you reconcile the answers to (b) and (c) above?(a) Find the equivalent surface and volume (bound) current densities J is permanently magnetized: The slab 7-14 M =a M, The regions z> are air. or bn b Find the magnetic field B everywhere. (c) Find H everywhere. J m' ms' (0