Question

A \(0.90-\mathrm{m}\) -long solenoid has a radius of \(5.0 \mathrm{~mm}\). When the wire carries a \(0.20-\mathrm{A}\) current, the magnetic field inside the solenoid is \(5.0 \mathrm{mT}\). How manv turns of wire are there in the solenoid?

Short answer

Answer: Approximately 17907 turns.

Step-by-step solution

Identify relevant quantities and formula

We are given: - Length of solenoid (L) = 0.90 m - Radius of solenoid (r) = 5.0 mm = 0.005 m - Current (I) = 0.20 A - Magnetic field inside solenoid (B) = 5.0 mT = 5.0 x 10^{-3} T We need to find the number of turns of wire (N) in the solenoid. Formula for magnetic field inside an ideal solenoid is given by: B = µ₀ * n * I where B is the magnetic field, µ₀ is the permeability of free space (µ₀ = 4π x 10^{-7} Tm/A), n is the number of turns per unit length (N/L) and I is the current. From this formula, we can derive the expression for N: N = (B * L) / (µ₀ * I)

Calculate the number of turns of wire

Now we substitute the given values into the formula: N = ((5.0 x 10^{-3} T) * (0.90 m)) / ((4π x 10^{-7} Tm/A) * (0.20 A)) Let's solve this equation step by step: N = (4.5 x 10^{-3}) / (8π x 10^{-8}) N = 4.5 / (8π x 10^{-5}) N = (4.5 / 8π) × 10^5 N ≈ 17907

Round the result

Since the number of turns of wire should be a whole number, we will round the result: N ≈ 17907 turns So, there are approximately 17907 turns of wire in the solenoid.

Key concepts

Magnetic Field

The magnetic field is a fundamental concept in physics, especially in the study of electromagnetism. A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A common source of a magnetic field is an electromagnet, which is created by wrapping a coil of wire (solenoid) around a magnetic core and passing an electric current through it.

A magnetic field can be visualized using magnetic field lines, which show the direction and strength of the field. The field is strongest where the lines are most densely packed, and the field's direction is the direction in which a North seeking pole of a magnet would point. This concept becomes particularly useful when calculating the properties of a solenoid, as we can use the uniform magnetic field created inside a solenoid to compute various parameters such as the number of wire turns necessary to produce a specific field strength.

Permeability of Free Space

The permeability of free space, represented by the symbol \( \mu_0 \), is a physical constant that is a measure of the ability of a vacuum to support the formation of a magnetic field. Its value is precisely \( 4\pi \times 10^{-7} \) Tesla meter per Ampere (Tm/A) by definition in the International System of Units (SI). This constant is crucial in the context of magnetic field calculations within a vacuum and also affects the behavior of magnetic fields in other materials. For example, the strength of the magnetic field \( B \) inside an ideal solenoid can be calculated using the formula \( B = \mu_0 \times n \times I \) where \( n \) is the number of turns per unit length and \( I \) is the electric current. Permeability is essential in understanding the characteristics of a magnetic field in different environments and is a key element in formulating equations in electromagnetism.

Turns Per Unit Length

When it comes to solenoids, turns per unit length (represented by \( n \) or sometimes \( N/L \) where \( N \) is the total number of turns and \( L \) is the length of the solenoid) is a critical measurement that gives us insight into the concentration of coil windings in a given length. It tells us how many turns of wire exist in one meter of the solenoid. Physically, more turns per unit length mean a higher density of wire, which usually results in a stronger magnetic field when a current is applied, provided that the other parameters such as current and the core material stay constant.

Toledo tailor the strength of the magnetic field produced by a solenoid, engineers and physicists adjust the number of wire turns it has, highlighting the importance of this measurement. The number of turns per unit length is used directly in the calculation of the magnetic field strength inside the solenoid and plays a pivotal role when solving problems related to electromagnetic devices.