Problem 82 - University Physics with Modern Physics

Question

To use a larger sample, the experimenters construct a solenoid that has the same length, type of wire, and loop spacing but twice the diameter of the original. How does the maximum possible magnetic torque on a bacterium in this new solenoid compare with the torque the bacterium would have experienced in the original solenoid? Assume that the currents in the solenoids are the same. The maximum torque in the new solenoid is (a) twice that in the original one; (b) half that in the original one; (c) the same as that in the original one; (d) one-quarter that in the original one.

Step-by-Step Solution

Answer

The maximum torque is (c) the same as that in the original one.

Step 1: Understanding Magnetic Torque

The magnetic torque \( \tau \) on a magnetic dipole is given by the formula \( \tau = \mu B \sin \theta \), where \( \mu \) is the magnetic moment, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the magnetic moment and the magnetic field. In this exercise, we seek to determine how the magnetic torque changes with a change in solenoid diameter.

Step 2: Influence of Solenoid Diameter on Magnetic Field

The magnetic field inside a long solenoid is given by \( B = \mu_0 \frac{nI}{L} \), where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, \( I \) is the current, and \( L \) is the length of the solenoid. If the diameter of the solenoid doubles, the cross-sectional area increases, but the magnetic field \( B \) is unaffected by the diameter and is determined only by the current and turn density.

Step 3: Evaluating Magnetic Torque with Changed Diameter

Since the magnetic moment \( \mu \) of a bacterium remains constant and the magnetic field \( B \) in the solenoid remains unchanged even with the doubled diameter, the magnetic torque \( \tau = \mu B \sin \theta \) experienced by the bacterium does not change.

Key Concepts

Solenoid

A solenoid is a long coil of wire wound in the form of a helix. When an electric current flows through the wire, it produces a magnetic field. Solenoids are crucial in many electrical and magnetic applications because they create a uniform magnetic field inside. The solenoid's length, type of wire, and loop spacing are consistent factors that contribute to the magnetic field's magnitude.
One essential characteristic of a solenoid is that its magnetic field does not depend on the diameter of the coil. You might ask why diameter doesn’t influence the magnetic field. This is because within a solenoid, magnetic field lines are concentrated and uniform, largely determined by the number of loops per unit length and the current passing through the coil. This uniformity is what makes solenoids valuable in applications involving magnetic fields.
In our exercise, even with a doubled diameter, the magnetic field produced by the solenoid remains unchanged if current and loop spacing stay constant.

Magnetic Dipole

A magnetic dipole refers to a system that has two magnetic poles - a north and a south pole. Think of a simple bar magnet or a small loop of wire with current. These dipoles create their own magnetic fields, which can exert force on each other under different conditions.
In this context, let's consider the magnetic dipole of a bacterium. Imagine a tiny magnet inside the bacterium, aligning itself with the magnetic field of the solenoid. These dipole moments can pivot and rotate when placed in external magnetic fields, which is how torques can be exerted on them. This torque influences how the bacterium or its embedded materials behave in the magnetic field created by the solenoid.
The dipole moment remains constant in different experimental setups unless the bacterium itself changes. Hence, this constant dipole moment is crucial when evaluating how different solenoids exert torque on it.

Magnetic Field Strength

Magnetic field strength refers to the intensity of a magnetic field in a region, denoted by the symbol \( B \). In a solenoid, this field strength is primarily determined by the current through the wire and the density of the coils, not by the solenoid's diameter.
Let's break down what affects \( B \):

• **Magnetic Permeability**: This is a constant (\( \mu_0 \)) of the material that shows how well it can carry magnetic fields.
• **Number of Turns per Unit Length**: Denoted as \( n \), more turns mean a stronger magnetic field.
• **Current \( I \)**: Higher current increases the field strength within the solenoid.

In the given scenario, these three factors remain unchanged despite the increased diameter of the solenoid. Thus, the magnetic field strength experienced by the bacterium remains the same, ensuring consistent measurement and influence of torque.

Magnetic Moment

The magnetic moment, often represented by the symbol \( \mu \), is a measure of the strength and direction of a magnetic source. This property is intrinsic to the object, such as a bacterium, and represents how it responds to a magnetic field.
When placed in a magnetic field, the magnetic moment of an object aligns to minimize potential energy—it acts like a compass needle wanting to align with Earth's magnetic field. The alignment affects the torque experienced by the object.
In the experiment, the bacterium's magnetic moment remains constant regardless of the solenoid's diameter. This constancy means that as long as the magnetic field strength \( B \) does not change, there is no variation in magnetic torque \( \tau \). Even with different solenoid dimensions, the magnetic moment ensures consistent behavior of the microbiological sample within the magnetic field.