Ch. 28 - Sources of Magnetic Field

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 28 - Sources of Magnetic Field

문제 38

27장, 문제 38

(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 $\vec{B}$ at the center of the ring?

검증된 단계별 안내

Understand the problem: The circular conducting ring has a current I flowing through it, splitting into unequal portions due to unequal resistance. We need to calculate the magnetic field B at the center of the ring caused by the current distribution.

Recall the Biot-Savart Law: The magnetic field at a point due to a current element is given by:

B = \frac{μ}{0} \frac{I}{4π} \frac{dL}{r} \sin θ

, where μ₀ is the permeability of free space, I is the current, dL is the length of the current element, r is the distance from the element to the point, and θ is the angle between dL and r.

Analyze the symmetry: The ring is circular, and the current flows through it symmetrically. At the center of the ring, contributions to the magnetic field from opposite sides of the ring will add up due to symmetry.

Set up the integral: To find the total magnetic field at the center, integrate the contributions from all current elements around the ring. The formula simplifies for a circular loop to:

B = \frac{μ}{0} \frac{I}{2R}

, where R is the radius of the ring.

Account for unequal current distribution: Since the current splits into unequal portions, calculate the magnetic field contributions from each segment separately using the Biot-Savart Law, and then sum them to find the net magnetic field at the center of the ring.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Magnetic Field Due to a Current-Carrying Loop

A circular conducting loop carrying a current generates a magnetic field in the space around it. The direction of the magnetic field at the center of the loop can be determined using the right-hand rule, where the thumb points in the direction of the current and the fingers curl in the direction of the magnetic field lines. The strength of the magnetic field at the center is given by the formula B = (μ₀I)/(2R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the loop.

Current Division in Parallel Circuits

When a current splits between two or more paths in a circuit, the division of current depends on the resistance of each path. According to Ohm's law, the current through a resistor is inversely proportional to its resistance. In this scenario, the unequal resistances of the wires connected to the ring will cause the current to divide unevenly, affecting the total current flowing through the ring and consequently the magnetic field produced.

Superposition of Magnetic Fields

The principle of superposition states that the total magnetic field at a point due to multiple sources is the vector sum of the individual magnetic fields produced by each source. In this case, the magnetic field at the center of the ring will be influenced by the contributions from both the current in the ring and the currents in the exterior wires. Understanding how to calculate the resultant magnetic field from these contributions is essential for solving the problem.

관련 실천

교과서 질문

(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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교과서 질문

(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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교과서 질문

(II) An electron enters a uniform magnetic field B = 0.28 T at a 45° angle to $\vec{B}$ . 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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교과서 질문

(II) Consider a straight section of wire of length d, as in Fig. 28–51, which carries a current I. (a) Show that the magnetic field at a point P a distance 𝑅 from the wire along its perpendicular bisector is

$B = \frac{μ_{0} I}{2 π R} \frac{d}{(d^{2} + 4 R^{2})^{\frac{1}{2}}}$

(b) Show that this is consistent with Example 28–10 for an infinite wire.

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교과서 질문

(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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교과서 질문

(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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(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?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Magnetic Field Due to a Current-Carrying Loop

Current Division in Parallel Circuits

Superposition of Magnetic Fields

(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 $\vec{B}$ at the center of the ring?