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Chapter 27: Problem 55

A straight wire carrying a current of 3.41 A is placed at an angle of \(10.0^{\circ}\) to the horizontal between the pole tips of a magnet producing a field of \(0.220 \mathrm{~T}\) upward. The poles tips each have a \(10.0 \mathrm{~cm}\) diameter. The magnetic force causes the wire to move out of the space between the poles. What is the magnitude of that force?

Short Answer

Expert verified
Answer: The magnitude of the force on the wire is approximately 0.129 N.

Step by step solution

01

Convert the length of wire to meters.

Given the diameter of the pole tips as 10 cm, we assume the length of wire under the influence of the magnetic field (L) is also 10 cm. To convert it into meters: \( L = \frac{10}{100} = 0.1 \: m\)
02

Convert the angle to radians.

The angle is given in degrees, but the trigonometric functions (__sin__ in this case) need the angle in radians. Convert the angle from degrees to radians using the formula: \(\textrm{Angle in radians} = \frac{\textrm{Angle in degrees} \cdot \pi}{180}\) So, \(\theta = \frac{10 \cdot \pi}{180} = \frac{\pi}{18}\mathrm{~radians}\).
03

Calculate the magnetic force on the wire.

Now, using the formula \(F = I \cdot L \cdot B \cdot sin(\theta)\), we can calculate the magnitude of the magnetic force on the wire: \(F = (3.41 \: A) \cdot (0.1 \: m) \cdot (0.220 \: T) \cdot sin(\frac{\pi}{18})\) Using a calculator to perform the arithmetic, we get: \(F \approx 0.129 \: N\) So the magnitude of the force on the wire is approximately 0.129 N.

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Key Concepts

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

Magnetic Field Strength
Magnetic field strength, often symbolized as B, is a quantitative expression of the intensity of a magnetic field at a particular location. Measured in teslas (T), it represents how strong the magnetic influence is that can exert a force on moving charges or current-carrying conductors.

In the given exercise, the magnetic field strength is provided as 0.220 T, indicating a relatively strong field capable of influencing nearby magnetic materials or electrical currents. When a wire with an electric current is placed within a magnetic field, this strength will directly impact the magnitude of the force that acts on it.
Current-Carrying Conductor
A current-carrying conductor, such as a wire, is a material that allows electric charge to flow through it. When electric current passes through a conductor and it is situated within a magnetic field, it experiences a force鈥攖his is the fundamental principle behind electric motors and generators.

In our scenario, the wire carries a current of 3.41 A, and due to this current, it will interact with the magnetic field between the poles of the magnet. The force on the wire depends on the current magnitude, the length of the wire within the magnetic field, and the orientation of the wire relative to the field.
Lorentz Force
The Lorentz force is the combined electric and magnetic force on a point charge due to electromagnetic fields. For a current-carrying wire, this force can be explained as the force exerted by a magnetic field B on a wire of length L carrying a current I. The magnitude of the force is given by the formula
\[ F = I \cdot L \cdot B \cdot \sin(\theta) \]

In this formula, \( \theta \) is the angle between the wire and the direction of the magnetic field. The sine function represents the component of the force perpendicular to the current flow, which is solely responsible for the motor effect 鈥� the physical movement of the wire in this context. The calculated force in our exercise is a direct application of the Lorentz force equation.
Angle Conversion Radians Degrees
Angles can be measured in both degrees and radians. While degrees are often more intuitive for people to understand, radians are the standard unit of angular measure used in mathematics, especially in trigonometry and calculus.

The conversion between degrees and radians is vital in solving physics problems involving angular measurements. One full revolution around a circle corresponds to \(360^\circ\) or \(2\pi\) radians. Therefore, to convert degrees to radians, we multiply by \(\frac{\pi}{180}\).
\[ \text{Radians} = \text{Degrees} \cdot \frac{\pi}{180} \]

Applying this to the exercise, the \(10.0^\circ\) angle is converted to radians to use in the Lorentz force formula, resulting in \(\theta = \frac{\pi}{18}\) radians.

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Most popular questions from this chapter

A particle with a charge of \(20.0 \mu \mathrm{C}\) moves along the \(x\) -axis with a speed of \(50.0 \mathrm{~m} / \mathrm{s}\). It enters a magnetic field given by \(\vec{B}=0.300 \hat{y}+0.700 \hat{z},\) in teslas. Determine the magnitude and the direction of the magnetic force on the particle.

A rail gun accelerates a projectile from rest by using the magnetic force on a current-carrying wire. The wire has radius \(r=5.1 \cdot 10^{-4} \mathrm{~m}\) and is made of copper having a density of \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\). The gun consists of rails of length \(L=1.0 \mathrm{~m}\) in a constant magnetic field of magnitude \(B=2.0 \mathrm{~T}\) oriented perpendicular to the plane defined by the rails. The wire forms an electrical connection across the rails at one end of the rails. When triggered, a current of \(1.00 \cdot 10^{4}\) A flows through the wire, which accelerates the wire along the rails. Calculate the final speed of the wire as it leaves the rails. (Neglect friction.)

It would be mathematically possible, for a region with zero current density, to define a scalar magnetic potential analogous to the electrostatic potential: \(V_{B}(\vec{r})=-\int_{\vec{r}_{0}}^{\vec{r}} \vec{B} \cdot d \vec{s},\) or \(\vec{B}(\vec{r})=-\nabla V_{B}(\vec{r}) .\) However, this has not been done. Explain why not.

A proton with an initial velocity given by \((1.0 \hat{x}+$$2.0 \hat{y}+3.0 \hat{z})\left(10^{5} \mathrm{~m} / \mathrm{s}\right)\) enters a magnetic field given by \((0.50 \mathrm{~T}) \hat{z}\). Describe the motion of the proton

A straight wire of length \(2.00 \mathrm{~m}\) carries a current of \(24.0 \mathrm{~A} .\) It is placed on a horizontal tabletop in a uniform horizontal magnetic field. The wire makes an angle of \(30.0^{\circ}\) with the magnetic field lines. If the magnitude of the force on the wire is \(0.500 \mathrm{~N}\), what is the magnitude of the magnetic field?
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