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Chapter 9: Problem 49

A straight section \(P Q\) of a circuit lies along the \(x\) -axis from \(x=-(a / 2)\) to \(x=+(a / 2)\) and carries a steady current 1 . The magnetic field due to the section \(P Q\) at a point \(x=+a\) will be a proportional to \(a\) b proportional to \(a^{2}\) c. proportional to \((1 / a)\) d equal to zero

Short Answer

Expert verified
The magnetic field at \(x = +a\) is zero due to symmetry.

Step by step solution

01

Understand the Scenario

The problem involves a straight current-carrying wire segment of length \(a\) along the \(x\)-axis, specifically from \(x = -\frac{a}{2}\) to \(x = +\frac{a}{2}\). We need to determine the magnetic field at a point located further along the \(x\)-axis at \(x = a\).
02

Use Biot-Savart Law

The Biot-Savart law is utilized to calculate the magnetic field due to a current element. It states that the magnetic field \(dB\) due to a small element of wire carrying current \(I\) is \(dB = \frac{{\mu_0}}{{4\pi}} \frac{{I d\mathbf{l} \times \mathbf{r}}}{{r^3}}\), where \(\mathbf{r}\) is the position vector from the element to the point of interest.
03

Consider Symmetry and Evaluate Magnetic Field

For a straight segment on the \(x\)-axis, calculate the magnetic field contributions. At points on the same axis, the perpendicular component of \(d\mathbf{B}\) becomes zero, due to symmetry around the line of current (no magnetic field along the axis of the wire itself). Outer points (on the axis) feel no field because all contributions cancel out symmetrically.
04

Conclude Based on Symmetry

Since exact straight-on axis positioning results in balanced, opposing magnetic fields due to each infinitesimal segment of the wire, the net magnetic field at \(x = +a\) from the segment is zero. This conclusion arises from all opposing contributions canceling each other out along a straight line.

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Key Concepts

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

Understanding Magnetic Fields
Magnetic fields are regions around a current-carrying wire where magnetic forces can be detected. They are invisible fields that exert a magnetic force on moving electric charges and magnetic dipoles. The strength and direction of a magnetic field depend on various factors, such as the magnitude of the current and the shape of the wire. Magnetic fields can be visualized using magnetic field lines, which emerge from the north pole of a magnet and enter the south pole. For a current-carrying wire, these lines form concentric circles around the wire. The closer you are to the wire, the stronger the magnetic field is, because the lines are closer together.

Biot-Savart Law helps us calculate these fields by taking into account the current element, the distance from the element to the point of interest, and the angle between the current element and the line connecting it to the point in question.
Exploring Current-Carrying Wires
A current-carrying wire generates a magnetic field around it. This happens because moving electric charges create magnetic fields. The direction of the magnetic field produced by a straight current-carrying wire can be determined by the right-hand rule. By pointing the thumb of your right hand in the direction of the electric current flow, your fingers will curl around the wire in the direction of the magnetic field lines.

The strength of a magnetic field due to a straight current-carrying wire decreases with distance from the wire. This is because magnetic field lines spread out as they move farther from the wire.
  • Magnetic fields created by current are integral in many technologies, like MRI machines and electric motors.
  • Understanding these fields is crucial for designing circuits and electrical devices.
Symmetry in Physics and Magnetic Fields
Symmetry is a fundamental concept in physics, often simplifying complex calculations. In the context of the magnetic field produced by a straight wire, symmetry helps us understand why certain positions experience no net magnetic field. Along the axis of a current-carrying wire, the magnetic field contributions from symmetrical positions around the wire cancel each other out. This is a direct consequence of the symmetry of the problem.

For instance, in the scenario of a wire on the x-axis, each tiny segment of the wire creates a magnetic field. However, due to symmetry, contributions from segments on opposite sides cancel each other, resulting in no net magnetic field at points directly along the wire's axis. This principle not only helps solve problems but also deepens our understanding of how physical laws are interconnected.
  • It illustrates how symmetric arrangements in physics often lead to simplifications in calculations.
  • Symmetry is a key principle in many areas of physics, including electromagnetism, quantum mechanics, and relativity.

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Most popular questions from this chapter

The value of the electric field strength in vacuum if the energy density is same as that due to a magnetic field of induction \(1 \mathrm{~T}\) in vacuum is a. \(3 \times 10^{8} \mathrm{NC}^{-1}\) b \(1.5 \times 10^{8} \mathrm{NC}^{-1}\) c. \(2.0 \times 10^{8} \mathrm{NC}^{-1}\) d. \(1.0 \times 10^{8} \mathrm{NC}^{-1}\)

An electron is moving along positive \(x\) -axis. A uniform electric field exists toward negative \(y\) -axis. What should be the direction of magnetic field of suitable magnitude so that net force on the electron is zero? a. Positive \(z\) -axis b Negative \(z\) -axis c. Positive \(y\) -axis d. Negative \(y\) -axis

The magnetic field due to a current carrying circular loop of radius \(3 \mathrm{~cm}\) at a point on the axis at a distance of \(4 \mathrm{~cm}\) from the center is \(54 \mathrm{mT}\). Its value at the center of the loop will be a \(250 \mu \mathrm{T}\) b \(150 \mu \mathrm{T}\) c. \(125 \mu \mathrm{T}\) d. \(75 \mu \mathrm{T}\)

A current \(l\) flows a thin wire shaped as regular polygon of \(n\) sides which can be inscribed in a circle of radius \(R\). The magnetic field induction at the center of polygon due to one side of the polygon is a \(\frac{\mu_{0} I}{\pi R}\left(\tan \frac{\pi}{n}\right)\) \& \(\frac{\mu_{0} I}{4 \pi R} \tan \frac{\pi}{n}\) c. \(\frac{\mu_{0} I}{2 \pi R}\left(\tan \frac{\pi}{n}\right)\) d \(\frac{\mu_{0} I}{2 \pi R}\left(\cos \frac{\pi}{n}\right)\)

A loop of flexible conducting wire of length \(\ell\) lies in magnetic field \(B\) which is normal to the plane of loop. A current \(I\) is passed through the loop. The tension developed in the wire to open up is a. \(\frac{\pi}{2} B I \ell\) b. \(\frac{B I \ell}{2}\) c. \(\frac{B l \ell}{2 \pi}\) d. BI\ell
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