Problem 14 - Applied Physics

Question

An earthmover requires a solenoid with \(25 \overline{0} 0\) turns and whose length is \(0.150 \mathrm{~m}\). What current is required to produce a magnetic field of \(0.100 \mathrm{~T}\) at the center of the solenoid?

Step-by-Step Solution

Answer

The required current is approximately 1.20 A.

Step 1: Understand the Formula for Magnetic Field of a Solenoid

The magnetic field at the center of a solenoid can be expressed with the formula \( B = \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space (approximately \( 4\pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m/A} \)), \( n \) is the number of turns per unit length of the solenoid, and \( I \) is the current.

Step 2: Calculate the Number of Turns per Unit Length

The total number of turns \( N \) is \( 25,000 \), and the length \( L \) of the solenoid is \( 0.150 \mathrm{~m} \). Calculate \( n = \frac{N}{L} = \frac{25,000}{0.150} \) turns per meter.

Step 3: Calculate the Required Current

Rearrange the formula \( B = \mu_0 \cdot n \cdot I \) to solve for current \( I \): \( I = \frac{B}{\mu_0 \cdot n} \). Substitute the known values: \( B = 0.100 \mathrm{~T} \), \( \mu_0 = 4\pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m/A} \), and \( n \) calculated in the previous step.

Step 4: Compute the Value of the Current

First, compute \( n = \frac{25,000}{0.150} = 166,666.67 \) turns/m. Now substitute to find \( I \): \( I = \frac{0.100}{4\pi \times 10^{-7} \times 166,666.67} \). Calculate to find the current required.

Key Concepts

Electromagnetism

Electromagnetism is a fundamental branch of physics that describes how electric currents create magnetic fields, and conversely, how changing magnetic fields generate electric currents. This natural phenomenon is governed by key principles and equations, one of which is Ampère's Law. It states that electric currents induce magnetic fields that loop around them. A practical application of electromagnetism is the solenoid, a type of electromagnetic coil. When current flows through a solenoid, it generates a magnetic field similar to a bar magnet, with a distinct north and south pole. The strength and direction of this magnetic field can be precisely controlled by adjusting the current.

Solenoid Calculations

Solenoids are widely used in electromagnetism to create controlled magnetic fields. When computing the magnetic field of a solenoid, we utilize the formula: \( B = \mu_0 \cdot n \cdot I \). Here, \( B \) represents the magnetic field strength within the solenoid, \( \mu_0 \) is the magnetic constant (approximately \( 4\pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m/A} \)), \( n \) is the number of turns per meter, and \( I \) is the current through the solenoid.

• To determine \( n \), divide the total number of turns (\( N \)) by the solenoid's length (\( L \)).
• The resulting value \( n \) is the density of turns along the solenoid.
• By knowing \( n \), you can easily calculate the magnetic field given any current \( I \).

Physics Problem Solving

Effectively solving physics problems often requires a systematic approach. Understanding the given information and what needs to be calculated is the first step. In the context of calculating the magnetic field of a solenoid:

• Identify your known values: total turns, solenoid length, and desired magnetic field strength.
• Apply the relevant formula and solve for the variable of interest, often requiring rearrangement of the equation.
• Break down the problem into simpler sub-steps, such as calculating the turns per meter \( n \) before finding the current \( I \).
• Double-check your calculations to ensure all units are consistent.

This structured process not only aids in arriving at the correct answer but helps in developing clear and logical thinking.

Current Calculation in Physics

The calculation of current is a vital step in many physics problems involving electromagnetism. In the case of a solenoid, determining the current required to achieve a specific magnetic field involves using the formula \( I = \frac{B}{\mu_0 \cdot n} \). Follow these steps:

• First, calculate the turns per meter (\( n \)) by dividing the total number of turns (\( N \)) by the solenoid's length (\( L \)).
• Substitute \( n \), \( B \), and \( \mu_0 \) into the current formula to find \( I \).
• Ensure all your calculations consider the correct values and units to avoid errors.

By systematically following these steps, you can effectively determine the current necessary to produce the desired magnetic field strength.