Does the Magnitude of Magnetic Field Change?

The magnitude of a magnetic field can change based on various factors, including the distance from the source, the strength of the current or magnet generating the field, and the presence of other magnetic materials or fields. Understanding the factors that influence the magnitude of a magnetic field is crucial in various applications, from magnetic measurements to nuclear magnetic resonance (NMR) phenomena.

Factors Affecting the Magnitude of Magnetic Field

Distance from the Source

The magnitude of a magnetic field decreases with the square of the distance from the source, as described by the inverse-square law. This relationship is expressed mathematically as:

$B = \frac{\mu_0 I}{2\pi r}$

where:
– $B$ is the magnetic field strength
– $\mu_0$ is the permeability of free space (4$\pi$ × 10^-7 T⋅m/A)
– $I$ is the current in the source
– $r$ is the distance from the source

This means that as the distance from the source increases, the magnetic field strength decreases rapidly. For example, doubling the distance from the source will result in a four-fold decrease in the magnetic field strength.

Strength of the Current or Magnet

The magnitude of the magnetic field is directly proportional to the strength of the current or the strength of the permanent magnet generating the field. This relationship is expressed by the following equations:

For a current-carrying wire:
$B = \frac{\mu_0 I}{2\pi r}$

For a permanent magnet:
$B = \frac{\mu_0 M}{4\pi}$

where:
– $M$ is the magnetization of the permanent magnet

Increasing the current or the strength of the permanent magnet will result in a corresponding increase in the magnitude of the magnetic field.

Presence of Magnetic Materials

The presence of magnetic materials, such as ferromagnetic or paramagnetic materials, can significantly affect the magnitude of the magnetic field. These materials can either enhance or distort the magnetic field, depending on their magnetic properties.

Ferromagnetic materials, like iron, nickel, and cobalt, can concentrate the magnetic field lines, leading to an increase in the local magnetic field strength. This effect is known as magnetic flux concentration and is commonly used in the design of transformers, electromagnets, and other magnetic devices.

Paramagnetic materials, on the other hand, can slightly enhance the magnetic field, while diamagnetic materials, such as copper and water, can slightly reduce the magnetic field.

Magnetic Measurements and Non-Uniformity

In the context of magnetic measurements, the uniformity of the magnetic field is crucial. Non-uniformity in the magnetic field can produce significant effects when making magnetic measurements on superconducting materials. This non-uniformity can lead to errors in the measurement of the magnetic moment of the sample, especially in systems where the field is changed while holding the sample stationary.

When the sample is subjected to a time-varying magnetic field, it can affect the magnetic history of the sample, leading to the motion of vortices, eddy currents, and other dynamic effects. These effects can introduce errors in the measurement of the magnetic moment, as the sample’s response to the changing field may not be instantaneous.

To account for the non-uniformity of the magnetic field, researchers often divide the wire’s length into N equal segments and measure the magnetic field values at equal intervals along the current’s path. The net force on the wire is then calculated as the sum of the forces on all these short segments.

Magnetic Field in NMR Phenomena

In NMR (Nuclear Magnetic Resonance) phenomena, the magnetic field plays a significant role in the number of measurable spin states (eigenstates) of a system. The number of observable spin states can be 0, ½, 1, or 2, depending on the strength of the external magnetic field.

When an unmagnetized sample of tissue is placed in an external magnetic field, net magnetization (M) develops, initially growing in the longitudinal direction. This growth is a simple exponential with a time constant T1, as the individual spins seek to align with the magnetic field.

Magnetization transfer is a process in which energy is transferred between macromolecular and free-water pools by irradiating the tissue with an off-resonance RF-pulse. This process can affect image contrast in NMR imaging, as it alters the magnetic properties of the tissue.

Examples and Numerical Problems

1. Example 1: Magnetic Field of a Current-Carrying Wire
2. Consider a current-carrying wire with a current of 5 A.

3. The magnetic field at a distance of 2 cm from the wire is given by:
   $B = \frac{\mu_0 I}{2\pi r} = \frac{4\pi \times 10^{-7} \text{ T⋅m/A} \times 5 \text{ A}}{2\pi \times 0.02 \text{ m}} = 1 \times 10^{-4} \text{ T}$

4. Example 2: Magnetic Field of a Permanent Magnet

5. Consider a permanent magnet with a magnetization of 1.2 T.

6. The magnetic field at a distance of 5 cm from the magnet is given by:
   $B = \frac{\mu_0 M}{4\pi} = \frac{4\pi \times 10^{-7} \text{ T⋅m/A} \times 1.2 \text{ T}}{4\pi} = 1.2 \times 10^{-4} \text{ T}$

7. Numerical Problem: Magnetic Force on a Current-Carrying Wire

8. A current-carrying wire with a length of 10 cm is placed in a non-uniform magnetic field.
9. The magnetic field values at different points along the wire are:
   • $B_1 = 0.5 \text{ T}$
   • $B_2 = 0.6 \text{ T}$
   • $B_3 = 0.7 \text{ T}$
10. The current in the wire is 2 A, and the angle between the current and the magnetic field is 30°.
11. Calculate the net force on the wire.

Solution:
– Divide the wire into 3 equal segments, each with a length of 3.33 cm.
– Calculate the force on each segment using the formula $F = ILBsin\theta$:
– Segment 1: $F_1 = 2 \text{ A} \times 0.0333 \text{ m} \times 0.5 \text{ T} \times sin(30°) = 0.0289 \text{ N}$
– Segment 2: $F_2 = 2 \text{ A} \times 0.0333 \text{ m} \times 0.6 \text{ T} \times sin(30°) = 0.0347 \text{ N}$
– Segment 3: $F_3 = 2 \text{ A} \times 0.0333 \text{ m} \times 0.7 \text{ T} \times sin(30°) = 0.0404 \text{ N}$
– The net force on the wire is the sum of the forces on the individual segments:
$F_\text{net} = F_1 + F_2 + F_3 = 0.0289 \text{ N} + 0.0347 \text{ N} + 0.0404 \text{ N} = 0.104 \text{ N}$

These examples and numerical problems demonstrate how the magnitude of the magnetic field can change based on various factors, such as distance, current/magnet strength, and the presence of magnetic materials. They also illustrate the importance of considering non-uniformity in the magnetic field when making accurate magnetic measurements.

References

1. Effects of Non-Uniform Magnetic Fields on Magnetic Measurements
2. Magnetic Force on a Current-Carrying Wire
3. NMR Phenomenon Quiz