The value of the line integral of B→⋅ds→\overrightarrow{B}\cdot\overrightarrow{ds} around the closed path in FIGURE EX29.21 is 1.38 x 10-5 T m. What are the direction (into or out of the figure) and magnitude of I3?

Step 1: Recall Ampere's Law, which states that the line integral of the magnetic field around a closed path is proportional to the net current enclosed by the path. The formula is: ∫→B⋅ds=μo⋅I, where μo is the permeability of free space and I is the net current enclosed. Step 2: Substitute the given value of the line integral into Ampere's Law. The line integral is given as 1.38×10-5⋅T⋅m. The permeability of free space is a constant: μo=4×10-7⋅T⋅m/A. Step 3: Rearrange Ampere's Law to solve for the net current enclosed, I. The formula becomes: I=∫→B⋅dsμo. Substitute the values for the line integral and μo into this equation. Step 4: Determine the direction of the current I3. Use the right-hand rule: if the magnetic field circulates counterclockwise around the closed path, the current is directed out of the figure. If the field circulates clockwise, the current is directed into the figure. Step 5: Calculate the magnitude of I3 using the rearranged formula from Step 3. Ensure the units are consistent and verify the direction using the right-hand rule.

Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Knight Calc 5th Edition

Ch 29: The Magnetic Field

문제 21

29장, 문제 21

The value of the line integral of $\vec{B} \cdot \vec{d s}$ around the closed path in FIGURE EX29.21 is 1.38 x 10^-5 T m. What are the direction (into or out of the figure) and magnitude of I₃?

검증된 단계별 안내

Step 1: Recall Ampere's Law, which states that the line integral of the magnetic field around a closed path is proportional to the net current enclosed by the path. The formula is:

\int_{\to B} \cdot d s = μ o \cdot I

, where

μ o

is the permeability of free space and

I

is the net current enclosed.

Step 2: Substitute the given value of the line integral into Ampere's Law. The line integral is given as

1.38 \times 10^{- 5} \cdot T \cdot m

. The permeability of free space is a constant:

μ o = 4 \times 10^{- 7} \cdot T \cdot m / A

Step 3: Rearrange Ampere's Law to solve for the net current enclosed,

I

. The formula becomes:

I = \frac{\int_{\to B} \cdot d s}{μ o}

. Substitute the values for the line integral and

μ o

into this equation.

Step 4: Determine the direction of the current

I_{3}

. Use the right-hand rule: if the magnetic field circulates counterclockwise around the closed path, the current is directed out of the figure. If the field circulates clockwise, the current is directed into the figure.

Step 5: Calculate the magnitude of

I_{3}

using the rearranged formula from Step 3. Ensure the units are consistent and verify the direction using the right-hand rule.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Line Integral of Magnetic Field

The line integral of a magnetic field (→B) along a closed path quantifies the total magnetic influence along that path. It is mathematically represented as ∮ →B ⋅ ds, where ds is an infinitesimal vector element of the path. This integral is crucial in applications of Ampère's Law, which relates the magnetic field around a closed loop to the electric current passing through the loop.

Ampère's Law

Ampère's Law states that the line integral of the magnetic field around a closed loop is proportional to the total current passing through the loop. Mathematically, it is expressed as ∮ →B ⋅ ds = μ₀I_enc, where I_enc is the enclosed current and μ₀ is the permeability of free space. This law is fundamental in electromagnetism, allowing us to relate magnetic fields to currents.

Direction of Current

The direction of current (I₃ in this case) can be determined using the right-hand rule, which states that if you curl the fingers of your right hand in the direction of the magnetic field lines, your thumb points in the direction of the current. Understanding the relationship between current direction and magnetic field orientation is essential for solving problems involving magnetic fields and currents, especially in closed loops.

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The value of the line integral of B→⋅ds→\(\overrightarrow{B}\[\cdot\]\overrightarrow{ds}\) around the closed path in FIGURE EX29.21 is 1.38 x 10-5 T m. What are the direction (into or out of the figure) and magnitude of I3?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Line Integral of Magnetic Field

Ampère's Law

Direction of Current

The value of the line integral of $\vec{B} \cdot \vec{d s}$ around the closed path in FIGURE EX29.21 is 1.38 x 10^-5 T m. What are the direction (into or out of the figure) and magnitude of I₃?