29. Sources of Magnetic Field

The single loop below has a radius of 10 cm and is perpendicular to the page (shown at a slight angle so you can better visualize it). If the magnetic field at the center is 10^-6 T directed left, what is the magnitude of the current? What is the direction of the current at the top of the wire:into the page (towards left) or out of the page (towards right)?

A long wire having total resistance of 10 Ω is made into a solenoid with 20 turns of wire per centimeter. The wire is connected to a battery, which provides a current in order to produce a 0.04 T magnetic field through the center of the solenoid. What voltage must this battery have?

The two tightly wound solenoids below both have length 40 cm and current 5 A in the directions shown. The left solenoid has radius 20 cm and 50 m of total wire. The right solenoid has radius 0.5 m and 314 m of total wire. The thinner solenoid is placed entirely inside the wider one so their central axes perfectly overlap. Assume wires don't touch. What is the magnitude and direction of the magnetic field that is produced by a combination of the two solenoids at their central axis?

(Note:your worksheet may have a typo and say "0.5 cm"for the right solenoid's radius; it should be 0.5 m.)

A thin 12-cm-long solenoid has a total of 370 turns of wire and carries a current of 2.0 A. Calculate the field inside the solenoid near the center."

(I) A 32-cm-long solenoid, 1.8 cm in diameter, is to produce a 0.30-T magnetic field at its center. If the maximum current is 5.4 A, how many turns must the solenoid have?

(II) A 650-turn horizontal solenoid is 15 cm long. The current in its loops is 38 A. A straight wire cuts through the center of the solenoid, along a 3.0-cm diameter. This wire carries a 22-A current downward (and is connected by other wires that don’t concern us). What is the force on this wire assuming the solenoid’s field points due east?

The magnetic field B at the center of a circular coil of wire carrying a current I (as in Fig. 27–9) is

B = ( μ₀NI ) / 2r

where N is the number of loops in the coil and r is its radius. Imagine a simple model in which the Earth’s magnetic field of about IG ( = 1 x 10⁻⁴ T) near the poles is produced by a single current loop around the equator. Roughly estimate the current this loop would carry.

(III) Start with the result of Example 28–11 for the magnetic field along the axis of a single circular loop of wire carrying a dc current I to obtain the field along the central axis inside a very long solenoid with n turns per meter (Eq. 28–4) that stretches from +∞ to −∞ ..

A small solenoid (radius rₐ ) is inside a larger solenoid (radius r₆ > rₐ ). They are coaxial with nₐ and n₆ turns per unit length, respectively. The solenoids carry the same current, but in opposite directions. Let r be the radial distance from the common axis of the solenoids. If the magnetic field inside the inner solenoid (r < rₐ) is to be in the opposite direction as the field between the solenoids (rₐ < r < r₆) , but have half the magnitude, determine the required ratio n₆/nₐ .

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius 𝑅 = 10.0 cm and are separated by a distance 𝑅 = 10.0 cm . Each coil has 85 loops carrying a current I = 2.0 A. (a) Determine the total magnetic field B along the 𝓍 axis (the center line for the two coils) in steps of 0.2 cm from the center of one coil (𝓍 = 0) to the center of the other (𝓍 = 𝑅 ).

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius 𝑅 = 10.0 cm and are separated by a distance 𝑅 = 10.0 cm . Each coil has 85 loops carrying a current I = 2.0 A. (c) By what % does B vary from 𝓍 = 5.0 cm to 𝓍 = 6.0 cm ?

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius 𝑅 = 10.0 cm and are separated by a distance 𝑅 = 10.0 cm . Each coil has 85 loops carrying a current I = 2.0 A. (b) Graph B as a function of 𝓍.

"(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 28–40. There is a dc current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

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(III) A square loop of wire, of side d, carries a current I. (a) Determine the magnetic field B at points on a line (call it the 𝓍 axis) perpendicular to the plane of the square which passes through the center of the square (Fig. 28–56). Express B as a function of 𝓍, the distance from the center of the square. (b) For 𝓍 ≫ d, does the square appear to be a magnetic dipole? If so, what is its dipole moment?

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Part of a long, thin insulated straight wire is formed into a single circular loop of radius 𝑅 (Fig. 28–68) and carries a current I. (a) What is the magnitude and direction of the magnetic field at the center of the loop? (b) If the plane of the loop is twisted 90 degrees so that the plane is perpendicular to the straight part of the wire (i.e., in the yz plane) what is the magnitude and direction of the field now at the center of the loop?

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(III) Use the result of Problem 44 to find the magnetic field at point P in Fig. 28–53 due to the current in the square loop.

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