Problem 59 - College Physics

Question

\(\bullet\) A closely wound circular coil has a radius of 6.00 \(\mathrm{cm}\) and carries a current of 2.50 A. How many turns must it have if the magnetic field at its center is \(6.39 \times 10^{-4} \mathrm{T} ?\)

Step-by-Step Solution

Answer

The coil must have 24 turns.

Step 1: Understand the Formula

The magnetic field at the center of a circular coil can be calculated using the formula: \(B = \frac{\mu_0 \cdot N \cdot I}{2R}\), where \(B\) is the magnetic field, \(N\) is the number of turns, \(I\) is the current, \(R\) is the radius, and \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \, \mathrm{T\cdot m/A})\).

Step 2: Substitute Known Values

Substitute the known values into the formula: \(B = 6.39 \times 10^{-4} \, \mathrm{T},\) \(R = 6.00 \, \mathrm{cm} = 0.06 \, \mathrm{m},\) and \(I = 2.50 \, \mathrm{A}.\) This gives the equation: \(6.39 \times 10^{-4} = \frac{4\pi \times 10^{-7} \cdot N \cdot 2.50}{2 \times 0.06}.\)

Step 3: Simplify the Equation

Simplify the equation to find \(N\). Start by multiplying both sides by \(2 \times 0.06\) to get \(7.668 \times 10^{-5}\). Then divide by \((4\pi \times 10^{-7} \times 2.50)\) to get \(N.\)

Step 4: Solve for N

Perform the calculations: first compute \(4\pi \times 10^{-7} \times 2.50 = 3.14 \times 10^{-6}.\) Then divide \(7.668 \times 10^{-5}\) by \(3.14 \times 10^{-6}\) to get \(N \approx 24.4.\)

Step 5: Interpret the Result

Since \(N\) must be an integer, and approximation leads to \(N = 24,\) the coil must have 24 turns to produce the specified magnetic field at its center.

Key Concepts

Understanding the Circular Coil

A circular coil is a wire looped multiple times in the form of a circle. It is a common component in electrical circuits, especially when dealing with magnetism.

The purpose of a circular coil is to generate a magnetic field when an electric current passes through it.

• The coil's configuration allows the superposition of magnetic fields from each turn, enhancing the overall magnetic effect at the center of the coil.
• Due to its circular shape, the field generated is more concentrated at the center compared to other parts of the coil.

This property is useful in many applications like electromagnets, inductors, and transformers, where a strong localized magnetic field is necessary.

Determining the Number of Turns

The number of turns in a coil is crucial in determining the strength of the magnetic field produced at the center.

When more turns are present:

• The magnetic field at the center increases proportionally.
• This is because each loop adds its magnetic field to the overall field.

Let's recall that in our formula, \[B = \frac{\mu_0 \cdot N \cdot I}{2R}\] - the term \(N\) represents these turns.
By rearranging the formula, we could solve for \(N\) if we know the desired magnetic field, radius, and current. This explains why it's vital to understand the number of turns when designing a coil for specific magnetic requirements.

Current in Coils and Its Effect on Magnetic Field

The current passing through the coil is another important factor. It directly influences the magnetic field strength within the coil.

According to the formula, the magnetic field \(B\) is directly proportional to the current \(I\):

• If the current increases, the magnetic field increases.
• Conversely, reducing the current decreases the magnetic field.

In practical terms, controlling the current can help manage the intensity of the magnetic field, making adjustments as needed for various applications. Understanding the relationship between current and magnetic field aids in designing efficient electromagnetic devices.

Permeability of Free Space

Permeability of free space, denoted as \(\mu_0\), is a physical constant that characterizes how a magnetic field interacts in a classical vacuum.

In our formulas involving circular coils, it allows us to predict the magnetic field strength given other factors like current and number of turns.

• The value of \(\mu_0\) is approximately \(4\pi \times 10^{-7} \, \mathrm{T \cdot m/A}\).
• It serves as a constant multiplier in equations to ensure the appropriate scale and unit conversion in calculations involving magnetic fields.

This constant highlights how essential it is to consider the medium through which a magnetic field extends, especially in theoretical and experimental physics.