Question

A toy airplane of mass \(0.175 \mathrm{~kg}\), with a charge of \(36 \mathrm{mC}\), is flying at a speed of \(2.8 \mathrm{~m} / \mathrm{s}\) at a height of \(17.2 \mathrm{~cm}\) above and parallel to a wire, which is carrying a 25 - A current; the airplane experiences some acceleration. Determine this acceleration.

Short answer

$$ B ≈ 1.83 × 10^{-5} \,\mathrm{T} $$ #tag_title#Step 2: Calculate the magnetic force acting on the airplane#tag_content#Next, we will calculate the magnetic force (F) acting on the airplane using the Lorentz force formula: $$ F = qvB \sin{\theta} $$ where \(F\) is the magnetic force, \(q\) is the electric charge of the airplane (1.5 µC = 1.5 × 10^{-6} C), \(v\) is the velocity of the airplane (65 m/s), \(B\) is the magnetic field strength (1.83 × 10^{-5} T), and \(\theta\) is the angle between the velocity and the magnetic field (90°). Since the angle is 90°, the sine value is 1, and the formula simplifies as: $$ F = qvB $$ Now, we can plug in the values and solve for \(F\): $$ F = (1.5 × 10^{-6}\,\mathrm{C})(65\,\mathrm{m/s})(1.83 × 10^{-5}\,\mathrm{T}) $$ $$ F ≈ 1.78 × 10^{-9}\,\mathrm{N} $$ #tag_title#Step 3: Determine the acceleration using Newton's second law#tag_content#Finally, we will determine the acceleration (a) of the toy airplane using Newton's second law of motion: $$ F = ma $$ where \(F\) is the magnetic force (1.78 × 10^{-9} N), \(m\) is the mass of the airplane (0.38 kg), and \(a\) is the acceleration. We need to solve for \(a\). Rearrange the formula to solve for \(a\): $$ a = \frac{F}{m} $$ Now, plug in the values and solve for \(a\): $$ a = \frac{1.78 × 10^{-9}\,\mathrm{N}}{0.38\,\mathrm{kg}} $$ $$ a ≈ 4.68 × 10^{-9}\,\mathrm{m/s^2} $$ #Answer# The acceleration experienced by the toy airplane due to the magnetic field produced by the wire is approximately \(4.68 × 10^{-9}\,\mathrm{m/s^2}\).

Step-by-step solution

Calculate the magnetic field produced by the wire

First, we will calculate the magnetic field (B) produced by the wire at the height of the airplane using the Biot-Savart law. The formula for the magnetic field produced by a long straight wire is $$ B = \frac{\mu_0 I}{2 \pi d} $$ where \(B\) is the magnetic field strength, \(\mu_0\) is the permeability of free space (\(4\pi × 10^{-7} \mathrm{Tm/A}\)), \(I\) is the current passing through the wire (25 A), and \(d\) is the distance from the wire to the airplane (17.2 cm = 0.172 m). Now, we can plug in the values and solve for \(B\): $$ B = \frac{4\pi × 10^{-7} \mathrm{Tm/A} × 25\,\mathrm{A}}{2 \pi × 0.172\,\mathrm{m}} $$

Key concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in magnetostatics that describes the magnetic field generated by steady electric currents. This law states that the magnetic field (\textbf{B}) at a point in space is directly proportional to the current (\textbf{I}) and inversely proportional to the distance (\textbf{d}) from the current-carrying conductor.

Mathematically, the law is expressed for a small segment of wire as:
\[ d\textbf{B} = \frac{\boldsymbol{\textbf{\textup{μ}}}_0}{4\textbf{\textup{π}}} \frac{I d\textbf{l} \times \boldsymbol{\textbf{\textup{r}}}}{r^3} \]
Here, \textbf{d\textbf{B}} is the infinitesimal contribution to the magnetic field from a small segment of wire with current \textbf{I}, \textbf{d\textbf{l}} is an infinitesimal length of wire, \textbf{r} is the position vector from the wire segment to the point in space, and \textbf{\textup{μ}}_0 is the permeability of free space. This equation underscores the vector nature of magnetic fields and their relationship to electric currents. Students can improve their understanding by visualizing how the current direction and the point's position relative to the wire influence the direction of the magnetic field using the right-hand rule.

Magnetic Field Strength

Magnetic field strength is a quantitative measure of the magnetic field's influence at a specific location in space, usually measured in Tesla (T). It is an essential concept for understanding how magnetic effects, such as those experienced by the toy airplane in our example, are calculated.

The strength of the magnetic field created by a long straight current-carrying conductor can be determined using the formula:
\[ B = \frac{\textbf{\textup{μ}}_0 I}{2\textbf{\textup{π}} d} \]
where \textbf{B} represents the magnetic field strength, \textbf{\textup{μ}}_0 signifies the permeability of free space, \textbf{I} is the current through the wire, and \textbf{d} is the perpendicular distance from the wire to the point of interest. Variations in any of these variables will alter the magnetic field's strength, influencing the force experienced by a nearby charged particle in motion.

Permeability of Free Space

The permeability of free space, denoted by \textbf{\textup{μ}}_0, is a physical constant that describes how a magnetic field can permeate the vacuum of free space. It is a critical factor in the formulas for calculating the magnetic field, acting as the proportionality constant in the Biot-Savart Law.

The value of \textbf{\textup{μ}}_0 is precisely: \[ \textbf{\textup{μ}}_0 = 4\textbf{\textup{π}} × 10^{-7}\thinspace\textup{Tm/A} \]This constant not only allows us to determine the strength of the magnetic field but also relates electric and magnetic phenomena, playing a key role in the equations of electromagnetism such as those formulated by Maxwell. When dealing with problems involving magnetic fields, such as the one involving the toy airplane, understanding how \textbf{\textup{μ}}_0 influences calculations is fundamental.

Force on a Charged Particle in a Magnetic Field

Understanding the force acting on a charged particle when it moves through a magnetic field is crucial for solving physics problems related to magnetism and electromagnetic interactions. The force (\textbf{F}) on a charged particle due to a magnetic field is given by the Lorentz force law, which states that the force is proportional to the charge (\textbf{q}), the velocity (\textbf{v}) of the particle, and the magnetic field (\textbf{B}) it is moving through.

The vector equation for the magnetic component of the Lorentz force is:
\[ \textbf{F} = \textbf{q} (\textbf{v} \times \textbf{B}) \]
The direction of this force is perpendicular to both the velocity of the particle and the magnetic field, which can be determined using the right-hand rule. In the scenario of the toy airplane, the charge, the airplane's velocity, and the strength and direction of the magnetic field from the current-carrying wire all interplay to determine the net force and consequent acceleration experienced by the airplane.