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Chapter 27: Problem 34

A straight, vertical wire carries a current of 2.60 A down- ward in a region between the poles of a large superconducting electromagnet, where the magnetic field has magnitude \(B=0.588 \mathrm{~T}\) and is horizontal. What are the magnitude and direction of the magnetic force on a \(1.00 \mathrm{~cm}\) section of the wire that is in this uniform magnetic field, if the magnetic field direction is (a) east; (b) south; (c) \(30.0^{\circ}\) south of west?

Short Answer

Expert verified
The magnitude of the force on the wire when the magnetic field is east is 0.0153 N towards the north, 0 when it is towards the south, and 0.0132 N north of east when it is \(30.0^{\circ}\) south of west.

Step by step solution

01

Identify given quantities

The current \(I\) is 2.60 A, the magnetic field strength \(B\) is 0.588 T, and the length of the wire \(L\) is 1.00 cm or 0.01 m (remember we should always convert cm to m).
02

Calculate the force for each given direction

We know from the formula \(F=BIL\sin(\theta)\) that the force is directly proportional to the sine of the angle between the direction of the current and the magnetic field direction. Since the current is downward, for (a) \(F=BIL\sin(90)\) because the magnetic field direction is east (perpendicular to the current), (b) \(F=BIL\sin(0)\) because the magnetic field is south (same direction as current). And for (c) since the current is downwards (south) and the magnetic field is \(30.0^{\circ}\) south of west, therefore the angle between them would be \(90+30=120^{\circ}\), so \(F=BIL\sin(120)\).
03

Compute the magnetic forces

Substitute the given values into the formula to compute the force in each case. For (a), the magnitude of the force \(F\) would be equal to (0.588 T) * (2.60 A) * (0.01 m) * \(\sin(90^{\circ}) = 0.0153 N\). Also, the direction of force would be to the north due to the right-hand rule. For (b), since the angle is 0^{\circ}, the force is 0. And for (c), the force will be (0.588 T) * (2.60 A) * (0.01 m) * \(\sin(120^{\circ}) = 0.0132 N\), and the direction will be north of east due to the right-hand rule.

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Key Concepts

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

Current-Carrying Wire
A current-carrying wire is one through which electric current flows. In this context, the wire carries a current of 2.60 A. The wire's orientation is vertical, meaning the current flows downward through it. Understanding current-carrying wires is crucial because they generate magnetic fields around them. This happens as electric charges move through the wire.
This scenario is a practical example of how this interaction plays out: a wire in the presence of a magnetic field experiences a force. This force's strength and direction depend on factors like:
  • The amount of current flowing through the wire
  • The length of the wire
  • The angle between the magnetic field lines and the wire
To solve problems regarding forces on a wire, we often use the formula for magnetic force: \[ F = BIL \sin(\theta) \] where \( F \) is the magnetic force, \( B \) the magnetic field strength, \( I \) the current, \( L \) the wire length, and \( \theta \) the angle between the current direction and the magnetic field lines.
Magnetic Field
Magnetic fields are invisible fields surrounding magnets or electric currents. In the exercise, the wire is placed in a magnetic field of 0.588 T (teslas), which is quite strong. A magnetic field exerts forces on moving charged particles, including those in current-carrying wires.
The magnetic field affects how the wire experiences forces when carrying an electric current. The force鈥檚 direction and magnitude can change based on the angle of the field relative to the wire. Causes of magnetic fields involve:
  • Permanent magnets that have a constant magnetic field
  • Electric currents creating a magnetic field around the conductor
  • Changing electric fields generating magnetic fields
Understanding magnetic fields involves recognizing the directions in which they act, which is often mapped using field lines. These lines indicate the path a north pole would take if placed in the field, flowing from north to south poles.
Right-Hand Rule
The right-hand rule is a mnemonic for determining the direction of the magnetic force on a current-carrying conductor in a magnetic field. It's a simple yet powerful tool used in electromagnetism. When faced with problems involving forces on a wire, the right-hand rule aids in visualizing the force direction.
To apply the right-hand rule:
  • Point your thumb in the direction of the current.
  • Position your fingers such that they point in the direction of the magnetic field.
  • Your palm or the force's resultant direction points perpendicular to both.
In the exercise, when calculating forces, this rule helps confirm the force directions: for instance, northward or northeastward, depending on the specific arrangement of current and magnetic field directions. It becomes crucial when figuring out whether the force is towards or away from specific points of the compass.
Electromagnetism
Electromagnetism is the branch of physics that studies electromagnetic forces. It's one of the fundamental forces of nature and describes interactions between electric charges and magnetic fields. When a wire carries a current, it interacts with nearby magnetic fields, demonstrating electromagnetic principles.
The scenarios described in the problem highlight concepts of electromagnetism such as:
  • The production of magnetic fields by electric currents
  • The interaction between currents and external magnetic fields
  • How this interaction results in forces acting on the current-carrying wire
Electromagnetic forces are calculated using the formula \( F = BIL\sin(\theta) \), which combines concepts of electric currents (from moving charges), magnetic fields (produced by these currents), and resulting forces. Such interactions form the basis for numerous technologies, including electric motors and transformers, where controlling electromagnetic forces is vital.

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