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Chapter 27: Problem 28

A conducting circular loop of radius \(r\) carries a constant current \(I\). It is placed in a uniform magnetic field \(B\) such that \(B_{0}\) is perpendicular to the plane of the loop. The magnetic force acting on the loop is: (a) \(\operatorname{Ir} B_{0}\) (b) \(2 \pi \operatorname{lr} B_{0}\) (c) \(\pi I r B_{0}\) (d) zero

Short Answer

Expert verified
(d) zero

Step by step solution

01

Understanding the Problem

We need to determine the magnetic force on a circular loop of radius \(r\) carrying a constant current \(I\), placed in a uniform magnetic field \(B_0\) that's perpendicular to the plane of the loop.
02

Lorentz Force Explanation

The Lorentz force on a current-carrying wire in a magnetic field is given by \( F = I \vec{L} \times \vec{B} \), where \( \vec{L} \) is the length vector of the wire and \( \vec{B} \) is the magnetic field. In this case, because the field is perpendicular to the plane of the loop, the forces will be symmetric around the loop.
03

Evaluating Symmetry

Since the magnetic field is uniform and normal to the plane of the loop, each small segment of the loop will experience a force that is radially outward. By symmetry, these forces cancel out, resulting in no net force on the entire loop.
04

Calculating the Force

To calculate the total force, consider a small segment of length \(dL\) of the circular loop. The force on this segment is \(dF = I dL B_0\) directed radially. The full loop can be considered as integrating these small segments around the circle, which due to symmetry results in net force being zero.
05

Conclusion Based on Calculation

Since the individual forces on each segment of the loop cancel out, the total magnetic force acting on the loop is zero.

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Key Concepts

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

Lorentz force
The Lorentz force is a fundamental concept in electromagnetism. It describes the force exerted on charged particles in the presence of electric and magnetic fields. For a straight wire carrying an electric current, the Lorentz force can be calculated using the formula:\[ F = I \vec{L} \times \vec{B} \]where:
  • \( F \) is the magnetic force
  • \( I \) is the current flowing through the wire
  • \( \vec{L} \) is the vector representing the length of the wire
  • \( \vec{B} \) is the magnetic field vector
In the context of a loop, the direction and magnitude of the force depend on the orientation of the loop in the magnetic field. The loop behaves as a collection of infinitesimally small segments, each experiencing a force. However, symmetry plays a crucial role in determining how these forces interact.
current-carrying loop
A current-carrying loop is essentially a closed circuit or a wire bent into a circular shape through which electrical current flows. When placed in a magnetic field, such loops can experience torque and forces. However, the situation changes when considering how the forces act on different parts of the loop. Each small segment of the loop experiences a magnetic force according to the Lorentz force law. In a circular loop, the situation is interesting because the forces on all segments work together towards a net effect:
  • Each segment feels a force perpendicular to the magnetic field and current direction.
  • Due to the loop's symmetry, these local forces can sum to different net results depending on the arrangement of the magnetic field.
In the specific case of a uniform field perpendicular to the plane of the loop, these forces balance out and cancel each other.
uniform magnetic field
A uniform magnetic field is one in which the magnetic field lines are parallel and equidistant from each other throughout the region. This means that the field has the same strength and direction everywhere within the specified area. When a current-carrying loop is placed in such a field, interesting phenomena occur:
  • The uniformity ensures that each segment of the loop encounters an equal magnetic influence.
  • This setup is ideal for assessing the net force because of how it simplifies the calculation of forces exerted on the loop.
  • A uniform magnetic field helps in demonstrating the principles of symmetry and force cancellation in a loop.
The field's uniform nature is crucial to the outcome in this exercise, where a net force of zero arises due to the symmetric distribution of the Lorentz force around the loop.
net force
Net force is the vector sum of all the forces acting on an object. In the case of a current-carrying loop placed in a uniform magnetic field, determining the net force involves analyzing forces acting on each segment of the loop. For the exercise given, here's how the concept applies:
  • Each segment of the loop experiences a force, calculated using the Lorentz force equation.
  • These forces act radially outward and are equal in magnitude due to the uniform field.
  • By symmetry, forces at opposite points in the loop cancel each other out.
Thus, despite individual forces acting on each segment, the sum of these forces results in a net force of zero. This cancellation of forces ensures that the loop remains undisturbed in its initial position. The understanding of net force is crucial for grasping how and why the total magnetic force on the loop is zero in this scenario.

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Most popular questions from this chapter

The strength of the magnetic field around a straight wire : (a) is same everywhere around the wire (b) obeys inverse square law (c) is directly proportional to the square of the distance from the wire (d) none of the above

If a charged particle is projected in a region of magnetic field, then : (a) the speed of the charged particle continuously changes (b) the magnetic force on the charged particle must be zero (c) the speed of the charged particle remains constant (d) the magnetic force on the charged particle must not be zero

If a charged particle of charge \(5 \mu \mathrm{C}\) and mass \(5 \mathrm{~g}\) is moving with constant speed \(5 \mathrm{~m} / \mathrm{s}\) in a uniform magnetic field \(B\) on a curve \(x^{2}+y^{2}=25\), where \(x\) and \(y\) are in metre. The value of magnetic field will be (a) 1 tesla (b) 1 kilo tesla along \(z\) -axis (c) 5 kilo tesla along the \(x\) -axis (d) 1 kilo tesla along any line in the \(x-y\) plane

A long wire carries a current of \(20 \mathrm{~A}\) along the axis of a solenoid, the field due to the solenoid is \(4 \mathrm{mT}\). The resultant field at a point \(3 \mathrm{~mm}\) from the solenoid axis is: (a) \(1.33 \mathrm{mT}\) (b) \(4.2 \mathrm{mT}\) (c) \(2.1 \mathrm{mT}\) (d) \(8.4 \mathrm{mT}\)

If the total magnetic field due to the earth is \(28 \mathrm{Am}^{-1}\), then the total magnetic induction due to the earth is : (a) \(3.52 \times 10^{-7} \mathrm{~T}\) (b) \(3.52 \times 10^{-5} \mathrm{~T}\) (c) \(3.52 \times 10^{-2} \mathrm{~T}\) (d) \(3.52 \times 10^{-4} \mathrm{~T}\)
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