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Mobile App Chapter 9: Problem 3
MAGNETIC FIELD CLOSE TO A CURRENT SHEET A conducting sheet carries a current density of \(\alpha\) amperes per meter. Show that, very close to the sheet, the magnetic induction \(B\) due to the current in the sheet is \(\mu_{0} \alpha / 2\) in the direction perpendicular to the current and parallel to the sheet.
Short Answer
Expert verified
The magnetic field close to the current sheet is \(\frac{\mu_0 \alpha}{2}\) perpendicular to the current and parallel to the sheet.
Step by step solution
01
Understand the use of Ampere's Law
Ampere's Law relates the integrated magnetic field along an imaginary closed loop, also known as the Amperian loop, to the electric current passing through that loop. The law is stated as \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\), where \(\vec{B}\) is the magnetic field, \(d\vec{l}\) is an infinitesimal vector element of the Amperian loop, and \(I_{enc}\) is the current enclosed by the Amperian loop.
02
Set up the Amperian Loop
Select an Amperian loop that is a rectangle, which extends above and below the current sheet. The length of the loop perpendicular to the current flow should be arbitrary, and its width, running parallel to the current and entering into the page, should also be arbitrary. This simplifies the integral on one side of the sheet.
03
Calculate Enclosed Current
The current enclosed by the Amperian loop is the current density multiplied by the width of the loop and the thickness of the current sheet. Because we're looking at the region very close to the sheet, the thickness of the Amperian loop can be considered infinitesimally small, and thus we may use the current density directly: \(I_{enc} = \alpha w\), where \(w\) is the width of the Amperian loop.
04
Apply Ampere's Law
Use Ampere's Law to calculate the magnetic field near the sheet. Given the symmetry of the problem, the magnetic field will only have components parallel to the sheet and will be the same on both sides of the loop. Thus, Ampere's Law simplifies to \(2Bl = \mu_0 \alpha w\), where \(l\) is the length of the Amperian loop.
05
Solve for the Magnetic Field
Solve for \(B\) by dividing both sides of the equation by \(2l\): \(B = \frac{\mu_0 \alpha w}{2l}\). Since \(w\) and \(l\) are arbitrary and cancel each other out, the resulting magnetic field close to the sheet is \(B = \frac{\mu_0 \alpha}{2}\).
06
State the Direction of the Magnetic Field
By the right-hand rule, the direction of the magnetic field lines will be perpendicular to the current density and parallel to the conducting sheet.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ampere's Law
Ampere's Law is a fundamental premise in electromagnetism, equating the magnetic field circulation around a closed loop to the total current passing through the loop. Mathematically, it is represented as \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\), with \(\vec{B}\) being the magnetic field, \(d\vec{l}\) a differential element of the loop's perimeter, and \(I_{enc}\) the current enclosed by the loop.
To apply this law to a current sheet, visualize an Amperian loop that strategically simplifies the calculation of the magnetic field. In the context of the original problem, we consider a loop that aligns perfectly with the uniform current sheet, ensuring that the field calculation only considers the current enclosed directly by the loop.
The beauty of Ampere's Law lies in its simplicity and broad applicability. It helps us derive the magnetic field in various scenarios, from long straight wires to complex current distributions. The law also implies that the magnetic field is a solenoidal vector field, exemplifying one of the four Maxwell's equations in integral form.
To apply this law to a current sheet, visualize an Amperian loop that strategically simplifies the calculation of the magnetic field. In the context of the original problem, we consider a loop that aligns perfectly with the uniform current sheet, ensuring that the field calculation only considers the current enclosed directly by the loop.
The beauty of Ampere's Law lies in its simplicity and broad applicability. It helps us derive the magnetic field in various scenarios, from long straight wires to complex current distributions. The law also implies that the magnetic field is a solenoidal vector field, exemplifying one of the four Maxwell's equations in integral form.
Current Density
Current density, symbolized as \(\alpha\), indicates the amount of electric current flowing per unit area of a cross-section. It is measured in amperes per meter (A/m). The versatility of current density allows it to describe not only currents in wires but also in continuous mediums like the current sheet mentioned in our exercise.
To understand it in a physical sense, imagine a highway with cars representing charges; the number of cars passing a certain point per second resembles the electric current, while the number of cars per lane width represents the current density.
Current density is central to calculating the magnetic field near the current sheet. Since the current is uniformly distributed across the sheet, the current density provides a way to determine the 'enclosed' current for any chosen width of the Amperian loop, a concept we apply directly in our original problem.
To understand it in a physical sense, imagine a highway with cars representing charges; the number of cars passing a certain point per second resembles the electric current, while the number of cars per lane width represents the current density.
Current density is central to calculating the magnetic field near the current sheet. Since the current is uniformly distributed across the sheet, the current density provides a way to determine the 'enclosed' current for any chosen width of the Amperian loop, a concept we apply directly in our original problem.
Magnetic Induction
Magnetic induction, often denoted as \(B\), represents the strength and direction of the magnetic field. In essence, it is the force felt by a moving charge or another magnetic object within a magnetic field. Its unit is the tesla (T) but can also be expressed in terms like \(\frac{Newton}{Ampere\cdot meter}\).
In the context of the described exercise, magnetic induction provides a quantitative measure of the influence that the current sheet has on its surroundings. The current sheet's magnetic induction \(B\) is directly proportional to the current density \(\alpha\), as evidenced by the derived formula \(B = \frac{\mu_0 \alpha}{2}\), showing a direct correlation between the two.
Understanding magnetic induction is essential not just for academic exercises but for real-world applications, such as in the design of electrical motors and generators, where the efficiency and performance can significantly depend on maintaining precise magnetic fields.
In the context of the described exercise, magnetic induction provides a quantitative measure of the influence that the current sheet has on its surroundings. The current sheet's magnetic induction \(B\) is directly proportional to the current density \(\alpha\), as evidenced by the derived formula \(B = \frac{\mu_0 \alpha}{2}\), showing a direct correlation between the two.
Understanding magnetic induction is essential not just for academic exercises but for real-world applications, such as in the design of electrical motors and generators, where the efficiency and performance can significantly depend on maintaining precise magnetic fields.
Amperian Loop
An Amperian loop is an imaginary closed loop used as a tool for applying Ampere's Law to calculate the magnetic field in a given situation. When selecting an Amperian loop, it is crucial to consider symmetry and the easiness to calculate the relevant components of the magnetic field.
In our example, the ideal Amperian loop is a rectangle that straddles the current sheet, because this shape captures the essence of the magnetic field produced without complicating the calculations. The strategic choice of loop, aligned with the direction of the current, ensures that the loop's contributions to the Ampere's integral result from either the entirety or none of the magnetic field, depending on the side.
Using an Amperian loop is an artistic science; it requires a blend of mathematical aptitude and physical insight. By carefully placing and shaping this loop around the current sheet, we create a bridge between a theoretical construct and a tangible measurement of the magnetic influence of the current.
In our example, the ideal Amperian loop is a rectangle that straddles the current sheet, because this shape captures the essence of the magnetic field produced without complicating the calculations. The strategic choice of loop, aligned with the direction of the current, ensures that the loop's contributions to the Ampere's integral result from either the entirety or none of the magnetic field, depending on the side.
Using an Amperian loop is an artistic science; it requires a blend of mathematical aptitude and physical insight. By carefully placing and shaping this loop around the current sheet, we create a bridge between a theoretical construct and a tangible measurement of the magnetic influence of the current.
