Problem 43 - University Physics with Modern Physics

Question

A magnetic field of 37.2 T has been achieved at the MTT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 \(\mathrm{cm}\) from a long, straight wire; \((b)\) at the center of a circular coil of radius 42.0 \(\mathrm{cm}\) that has 100 turns; \((\mathrm{c})\) near the center of a solenoid with radius \(2.40 \mathrm{cm},\) length \(32.0 \mathrm{cm},\) and \(40,000\) turns.

Step-by-Step Solution

Answer

(a) 1.86 × 10⁶ A, (b) 2.48 × 10⁶ A, (c) 9.47 A.

Step 1: Magnetic field from a long, straight wire

For part (a), the magnetic field created by a long, straight current-carrying wire can be calculated using the formula: \[ B = \frac{\mu_0 I}{2\pi r} \]where \( B = 37.2 \) T (the given magnetic field), \( \mu_0 = 4\pi \times 10^{-7} \) T·m/A (permeability of free space), and \( r = 2.00 \) cm or \( 0.0200 \) m (distance from the wire). Rearranging the formula to solve for the current \( I \), we have:\[ I = \frac{2\pi rB}{\mu_0} \]Substitute the known values:\[ I = \frac{2\pi \times 0.0200 \times 37.2}{4\pi \times 10^{-7}} \]\[ I = \frac{0.744}{4\times 10^{-7}} \]\[ I = 1.86 \times 10^{6} \] A.

Step 2: Magnetic field at the center of a circular coil

For part (b), the magnetic field at the center of a circular coil can be calculated using the formula:\[ B = \frac{\mu_0 N I}{2R} \]where \( R = 42.0 \) cm or \( 0.420 \) m (radius of the coil), \( N = 100 \) (number of turns), and \( B = 37.2 \) T. Rearranging this to solve for the current \( I \), we find:\[ I = \frac{2BR}{\mu_0 N} \]Substitute the values into the formula:\[ I = \frac{2 \times 37.2 \times 0.420}{4\pi \times 10^{-7} \times 100} \]\[ I = \frac{31.248}{4\pi \times 10^{-5}} \]\[ I = 2.48 \times 10^{6} \] A.

Step 3: Magnetic field near the center of a solenoid

For part (c), the magnetic field inside a solenoid is given by:\[ B = \frac{\mu_0 n I}{L} \]where \( n = \frac{N}{L} = \frac{40,000}{0.320} \) turns/m and \( B = 37.2 \) T. Substitute these values and \( L = 0.320 \) m into the formula to find \( I \):\[ I = \frac{BL}{\mu_0 n} \] Calculate \( n \):\[ n = \frac{40,000}{0.320} = 125,000 \] turns/m.Now calculate \( I \):\[ I = \frac{37.2 \times 0.320}{4\pi \times 10^{-7} \times 125,000} \]\[ I = \frac{11.904}{4\pi \times 10^{-2}} \]\[ I = 9.47 \] A.

Key Concepts

Magnetic Field from a Wire

Understanding the magnetic field generated by a current-carrying wire is crucial in electromagnetism. When a current flows through a long, straight wire, it creates a magnetic field around it. This magnetic field is described by Ampère's Law and calculated using the formula: \[ B = \frac{\mu_0 I}{2\pi r} \]where \( B \) represents the magnetic field, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \) T·m/A), \( I \) is the current, and \( r \) is the distance from the wire. This formula tells us that the magnetic field strength decreases with increasing distance from the wire.
When solving problems involving magnetic fields from wires, it is important to rearrange the formula to find the current when given other parameters.

• Magnetic field is directly proportional to the current \( I \).
• It is inversely proportional to the distance \( r \).
• A strong magnetic field requires a high current.

By applying these concepts, you can determine the necessary current to achieve a specific magnetic field at a given distance from the wire.

Magnetic Field of a Circular Coil

Circular coils are commonly used to generate magnetic fields for applications such as electromagnets and inductors. The magnetic field at the center of a circular coil depends on several factors, including the number of turns, the current flowing through the coil, and the coil's radius. The formula used to calculate this magnetic field is:\[ B = \frac{\mu_0 N I}{2R} \]where \( N \) is the number of turns, \( I \) is the current through the coil, and \( R \) is the radius of the coil. This relationship shows us that the magnetic field at the center:

• Increases with more turns \( N \) in the coil.
• Grows stronger with more current \( I \).
• Weakens with a larger radius \( R \).

In practical scenarios, to achieve a high magnetic field at the center, engineers might increase the number of turns or boost the current, keeping the coil compact. The interplay among these parameters is essential to tailor the desired magnetic field strength.

Magnetic Field inside a Solenoid

A solenoid is a long coil of wire, typically used to create a uniform magnetic field in a controlled space. The strength of the magnetic field inside a solenoid is distinguished from that of a simple wire or a circular coil due to its uniformity and confinement mostly within the solenoid. The expression for this magnetic field is given by:\[ B = \frac{\mu_0 n I}{L} \] where \( n \) is the number of turns per unit length, and \( L \) is the solenoid's length. This equation emphasizes that the magnetic field:

• Is proportional to the number of turns per meter \( n \).
• Depends directly on the current \( I \).
• Is largely independent of the solenoid's circumference or radius.

When calculating the magnetic field, it's essential to determine \( n \) as the number of turns divided by the solenoid’s length. Solenoids allow precise control over magnetic fields, making them valuable in scientific and industrial applications.