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Chapter 12: Problem 27

A certain region of space has a magnetic field given by \(\mathbf{B}=b x \hat{\mathbf{y}}\). Find the electric current flowing through the square defined by \(z=0,0 \leq x \leq a\), and \(0 \leq y \leq a\).(answer check available at lightandmatter.com)

Short Answer

Expert verified
The electric current flowing through the square is zero.

Step by step solution

01

Understand the Relationship

According to Amp猫re's Circuital Law and Maxwell's equations, a magnetic field can induce an electric current. Here, we need to understand how the magnetic field \(\mathbf{B}=b x \hat{\mathbf{y}}\) affects a square region in the xy-plane at \(z=0\), extending from \(0\) to \(a\) along both \(x\) and \(y\) directions.
02

Apply Amp猫re's Law

Ampere's law for current density states the integral of the current density over an area equals the line integral of the magnetic field over its boundary. We use:\[\oint_C \mathbf{B} \, \cdot \, d\mathbf{l} = \mu_0 I_{enclosed}\]where \(C\) is the boundary of the square and \(I_{enclosed}\) is the enclosed current.
03

Evaluate the Line Integral of \(\mathbf{B}\)

For our square path, the line integral of \(\mathbf{B}\) will be evaluated using the path segments along \(x=a,\ x=0,\ y=a,\ y=0\). Since \(\mathbf{B}\) is along the \(\hat{\mathbf{y}}\)-direction:- Along \(x = 0\) and \(x = a\), \(\mathbf{B} = 0\ no \text{ component parallel to } d\mathbf{l}\).- Along \(y=0\): \(\int_0^a b x \, dx \), yielding \(\frac{1}{2}abx^2 |_{0}^{a} = \frac{1}{2}a^2b\).- Along \(y=a\): subtraction for opposite direction, yielding \(- \frac{1}{2} a^2 b\).Summing gives: \(\oint_C \mathbf{B} \, \cdot \, d\mathbf{l} = 0\).
04

Solve for Current

In this case, because the line integral evaluated to zero, \(I_{enclosed} = 0\). Thus, no net electric current flows through the square. The symmetry and the specified boundary conditions contribute to this outcome.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field
A magnetic field is a vector field that represents the magnetic influence on moving electric charges, electric currents, and magnetic materials. In simplest terms, it's what magnets produce and affects other magnets or conductive materials in its vicinity. Magnetic fields have direction and magnitude, just like forces.
In the given exercise, the magnetic field \(\mathbf{B}=b x \hat{\mathbf{y}}\) is dependent on the variable \(x\), meaning it varies with position along the x-axis and points in the direction of the y-axis. This type of magnetic field can create complex interactions with currents and can change over space.
Understanding magnetic fields is crucial as they affect how currents move and are fundamental to electromagnetism.
Electric Current
Electric current refers to the flow of electric charge, typically measured in amperes. It is one of the essential aspects of electromagnetism, affecting and being affected by magnetic fields. When electric charges move, they generate a magnetic field. Conversely, moving through or in the presence of a magnetic field can influence an electric current.
In the problem, we explore whether the magnetic field induces a net electric current around a closed path. Despite the theoretical setup suggesting an interaction, the calculations show that no net current flows through the specified square area. This is a balance resulting from the symmetry of the magnetic field and the path chosen for evaluation.
Remember, current only flows if there's a potential difference and a closed loop for it to travel through, absent in the zero sum of enclosed current here.
Maxwell's Equations
Maxwell's equations are a set of four fundamental equations governing electromagnetism. They describe how electric and magnetic fields interact and how they relate to electric charges and currents. These equations unify the concepts of electricity, magnetism, and electromagnetism.
In this exercise, Amp猫re's law (one of Maxwell's equations) is applied. It connects the magnetic field circulating around a closed loop with the electric current passing through it:
  • \(\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enclosed}\)
This equation shows that for a magnetic field to induce an electric current, that current must be enclosed by the path through which the magnetic field circulates. However, in situations like our exercise where the integral resolves to zero, it illustrates that the magnetic field is balanced by the geometry of the setup, leading to no net current.
Maxwell's equations are the cornerstone of understanding how magnetic fields can generate forces and influence currents in electrical engineering and physics.

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Most popular questions from this chapter

Two parallel wires of length \(L\) carry currents \(I_{1}\) and \(I_{2}\). They are separated by a distance \(R\), and we assume \(R\) is much less than \(L\), so that our results for long, straight wires are accurate. The goal of this problem is to compute the magnetic forces acting between the wires. (a) Neither wire can make a force on itself. Therefore, our first step in computing wire 1's force on wire 2 is to find the magnetic field made only by wire 1 , in the space occupied by wire 2 . Express this field in terms of the given quantities.(answer check available at lightandmatter.com) (b) Let's model the current in wire 2 by pretending that there is a line charge inside it, possessing density per unit length \(\lambda_{2}\) and moving at velocity \(v_{2}\). Relate \(\lambda_{2}\) and \(v_{2}\) to the current \(I_{2}\), using the result of problem 5a. Now find the magnetic force wire 1 makes on wire 2, in terms of \(I_{1}, I_{2}, L\), and \(R\). Wwans \\{hwans:forcebetweentwowires\\} (c) Show that the units of the answer to part b work out to be newtons.

Electromagnetic waves are supposed to have their electric and magnetic fields perpendicular to each other. (Throughout this problem, assume we're talking about waves traveling through a vacuum, and that there is only a single sine wave traveling in a single direction, not a superposition of sine waves passing through each other.) Suppose someone claims they can make an electromagnetic wave in which the electric and magnetic fields lie in the same plane. Prove that this is impossible based on Maxwell's equations.

Two long, parallel strips of thin metal foil form a configuration like a long, narrow sandwich. The air gap between them has height \(h\), the width of each strip is \(w\), and their length is \(\ell\). Each strip carries current \(I\), and we assume for concreteness that the currents are in opposite directions, so that the magnetic force, \(F\), between the strips is repulsive. (a) Find the force in the limit of \(w \gg h\).(answer check available at lightandmatter.com) (b) Find the force in the limit of \(w \ll h\), which is like two ordinary wires. (c) Discuss the relationship between the two results.

A charged particle of mass \(m\) and charge \(q\) moves in a circle due to a uniform magnetic field of magnitude \(B\), which points perpendicular to the plane of the circle. a. Assume the particle is positively charged. Make a sketch showing the direction of motion and the direction of the field, and show that the resulting force is in the right direction to produce circular motion. b. Find the radius, \(r\), of the circle, in terms of \(m\), \(q\), \(v\), and \(B\).(answer check available at lightandmatter.com) c. Show that your result from part b has the right units. d. Discuss all four variables occurring on the right-hand side of your answer from part b. Do they make sense? For instance, what should happen to the radius when the magnetic field is made stronger? Does your equation behave this way? e. Restate your result so that it gives the particle's angular frequency, \(\omega\), in terms of the other variables, and show that \(v\) drops out.(answer check available at lightandmatter.com)

If you put four times more current through a solenoid, how many times more energy is stored in its magnetic field?(answer check available at lightandmatter.com)
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