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Ch. 28 - Sources of Magnetic Field
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 28 - Sources of Magnetic FieldProblem 33
Chapter 27, Problem 33

(II) An electron enters a uniform magnetic field B = 0.28 T at a 45° angle to B→\(\overrightarrow{B}\). Determine the radius r and pitch p (distance between loops) of the electron’s helical path assuming its speed is 2.2 x 106 m/s. See Fig. 27–48.


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Step 1: Break down the velocity of the electron into two components: one parallel to the magnetic field (v_parallel) and one perpendicular to the magnetic field (v_perpendicular). Use trigonometric functions: v_parallel = v * cos(θ) and v_perpendicular = v * sin(θ), where θ = 45° and v = 2.2 × 10⁶ m/s.
Step 2: Calculate the radius of the circular motion caused by the perpendicular velocity component. Use the formula r = (m * v_perpendicular) / (q * B), where m is the mass of the electron (9.11 × 10⁻³¹ kg), q is the charge of the electron (1.6 × 10⁻¹⁹ C), B is the magnetic field strength (0.28 T), and v_perpendicular is the velocity component perpendicular to the field.
Step 3: Determine the pitch of the helical path, which is the distance between successive loops. The pitch is given by p = v_parallel * T, where T is the period of the circular motion. First, calculate T using T = (2π * m) / (q * B). Then multiply T by v_parallel to find the pitch.
Step 4: Combine the results to describe the helical motion. The radius r determines the size of the circular motion, while the pitch p determines the spacing between loops along the direction of the magnetic field.
Step 5: Verify the units and ensure all calculations are consistent with SI units. This step ensures the physical interpretation of the radius and pitch aligns with the problem's context.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field. It is given by the equation F = q(v × B), where F is the force, q is the charge of the particle, v is its velocity, and B is the magnetic field. This force is perpendicular to both the velocity of the particle and the direction of the magnetic field, causing the particle to move in a circular or helical path.

Radius of Circular Motion

The radius of the circular motion of a charged particle in a magnetic field can be determined using the formula r = mv/(qB), where m is the mass of the particle, v is its speed, q is its charge, and B is the magnetic field strength. This radius indicates how tightly the particle spirals around the magnetic field lines, with a larger radius corresponding to lower magnetic forces or higher speeds.

Helical Motion

Helical motion occurs when a charged particle moves in a magnetic field at an angle, resulting in a spiral path. The pitch of the helix, which is the distance between successive loops, can be calculated using the formula p = v_parallel * T, where v_parallel is the component of the velocity parallel to the magnetic field and T is the period of the circular motion. This motion combines circular motion in the plane perpendicular to the magnetic field and linear motion along the field direction.
Related Practice
Textbook Question

(II) Two long parallel wires 8.20 cm apart carry 19.5-A dc currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other. See Fig. 28–43. [Hint: Use the law of cosines. See Appendix A or inside rear cover.]

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Textbook Question

(II) A circular conducting ring of radius 𝑅 is connected to two exterior straight wires at two ends of a diameter (Fig. 28–47). The current I splits into unequal portions as shown (unequal resistance) while passing through the ring. What is B→\(\overrightarrow{B}\) at the center of the ring?


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Textbook Question

(II) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 28–42). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 18.0 A and the bottom one carries 12.0 A?

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Textbook Question

(II) A wire is formed into the shape of two half circles connected by equal-length straight sections as shown in Fig. 28–48. A current I flows in the circuit clockwise as shown. Determine (a) the magnitude and direction of the magnetic field at the center, C, and (b) the magnetic dipole moment of the circuit.

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Textbook Question

(II) Let two long parallel wires, a distance d apart, carry equal dc currents I in the same direction. One wire is at 𝓍 = 0, the other at 𝓍 = d, Fig. 28–41. Determine B→\(\overrightarrow{B}\) along the 𝓍 axis between the wires as a function of 𝓍.

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Textbook Question

(III) A coaxial cable consists of a solid inner conductor of radius R1, surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3 (Fig. 28–45). The conductors carry equal and opposite currents I₀ distributed uniformly across their cross sections. Determine the magnetic field at a distance R from the axis for: (a) R < R1; (b) R1 < R < R2; (c) R2 < R < R3; (d) R > R3. (e) Let I₀ = 1.50 A, R1 = 1.00 cm , R2 = 2.00 cm , and R3 = 2.50 cm Graph B from R = 0 to R = 3.00 cm.

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