Question

The number of turns in a solenoid is doubled, and its length is halved. How does its magnetic field change? a) It doubles. c) It quadruples. b) It is halved. d) It remains unchanged.

Short answer

Answer: c) It quadruples.

Step-by-step solution

Recall the formula for the magnetic field inside a solenoid

The formula for the magnetic field (B) inside a solenoid of length (l) and number of turns (n) is given by: B = μ₀ * n * i Where μ₀ is the permeability of free space (constant) and i is the current passing through the solenoid. However, we need to consider the number of turns per unit length (n') which is given by: n' = n / l So, we can rewrite the formula for the magnetic field as: B = μ₀ * n' * i

Analyze the change in solenoid parameters

Now, let's analyze how the given changes in the exercise affect the solenoid parameters: 1. The number of turns (n) is doubled, so n_new = 2n. 2. The length (l) is halved, so l_new = 0.5l. We need to find the new magnetic field (B_new) inside the solenoid using the new number of turns per unit length (n_new').

Find n_new' - the new turns per unit length

We can find n_new' using the new number of turns (n_new) and the new length (l_new): n_new' = n_new / l_new n_new' = (2n) / (0.5l) n_new' = 4n / l Notice that n_new' = 4 * n', which is four times the initial turns per unit length of the solenoid.

Calculate the new magnetic field (B_new)

Now, we substitute n_new' into the magnetic field formula to find B_new: B_new = μ₀ * n_new' * i B_new = μ₀ * (4n') * i Since the initial magnetic field (B) is given by B = μ₀ * n' * i, we can write: B_new = 4 * (μ₀ * n' * i) B_new = 4 * B

Compare the initial and new magnetic fields

As we can see from the calculations, the new magnetic field (B_new) is four times the initial magnetic field (B): B_new = 4 * B So, the correct answer is: c) It quadruples.

Key concepts

Solenoid

A solenoid is a type of electromagnet, crucial to understanding the principles of electromagnetism and its practical applications. At its core, a solenoid consists of a coil of wire that is tightly wound in the shape of a helix. When an electric current flows through the coil, it generates a magnetic field within the interior of the solenoid.

The strength of this magnetic field depends on several factors: the number of coil turns, the current passing through the coil, and the properties of the space inside the coil. If the coil is wound around a ferromagnetic core, such as iron, this significantly enhances the magnetic field strength compared to a solenoid with an air core.

A solenoid acts very much like a bar magnet, with a north and a south pole. It produces a fairly uniform magnetic field in its interior, which is a unique characteristic that makes solenoids extremely useful in scientific experiments and various technological devices, from car starters to electromechanical switches and MRI machines.

Permeability of Free Space

The permeability of free space, often denoted as \(\boldsymbol{\mu_0}\), is a fundamental physical constant that characterizes the magnetic properties of the vacuum. It essentially quantifies how easily a magnetic field can penetrate the vacuum, or more generally, free space.

The permeability of free space is a crucial parameter in the equations that describe the relationship between magnetic fields and electrical currents, most notably in Ampere's Law and the Biot-Savart Law. The exact value of \(\mu_0\) is precisely defined and standardized, making it a cornerstone for magnetic field calculations in a vacuum.

It's useful to note that the permeability of a material is related, but not identical, to \(\mu_0\). Materials with permeability higher than \(\mu_0\) are considered ferromagnetic and can greatly enhance magnetic field strength when placed inside the coil of a solenoid, as opposed to non-magnetic materials which have permeability values close to \(\mu_0\).

Turns Per Unit Length

The concept of 'turns per unit length' in a solenoid measures how densely the coils are wrapped over a given length of the solenoid. Mathematically, it is denoted as \(n'\), and is calculated by dividing the total number of turns of the coil (\(n\)) by the length of the solenoid (\(l\)): \(n' = \frac{n}{l}\).

Turns per unit length is directly proportional to the magnetic field strength inside the solenoid. The more turns there are in a given length, the stronger the magnetic field, due to the additive nature of the magnetic field produced by each turn. Hence, if we change the number of turns or alter the length of the solenoid, this has a direct and predictable effect on the magnetic field produced, as observed in our exercise.

Engineers and physicists manipulate the turns per unit length to control the strength and uniformity of the magnetic field for a wide range of applications, from the precise readings needed in medical equipment to the raw power exerted by industrial electromagnets.

Ampere's Law

Ampere's Law is a foundational law in electromagnetism, named after André-Marie Ampère, which relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is one of Maxwell's four equations, which are the basis of classical electrodynamics.

Ampere's Law is mathematically expressed as \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\), where \(\vec{B}\) is the magnetic field, \(d\vec{l}\) is an infinitesimal element of the closed loop, \(\mu_0\) is the permeability of free space, and \(I_{enc}\) is the current enclosed by the loop.

In the context of a solenoid, Ampere's Law helps us understand why changing the number of turns or the length of the solenoid affects the magnetic field. By analyzing how the current and the turns per unit length contribute to the line integral of the magnetic field, we can calculate the total magnetic field within the solenoid, as demonstrated in the step-by-step textbook solution regarding the doubling of turns and the halving of the length of a solenoid.