Magnetic Field vs Magnetic Induction: A Comprehensive Guide

Magnetic field and magnetic induction are two closely related concepts in electromagnetism, but they have distinct physical meanings and units. The magnetic field, denoted as $\mathbf{H}$, is a measure of the magnetic force per unit charge, while magnetic induction, denoted as $\mathbf{B}$, is a measure of the magnetic flux density. Understanding the differences and relationships between these two quantities is crucial for understanding various electromagnetic phenomena and applications.

Magnetic Field ($\mathbf{H}$)

The magnetic field, $\mathbf{H}$, is a vector field that describes the magnetic force per unit charge at every point in space. It is measured in amperes per meter (A/m) in the SI system. The magnetic field is often generated by moving charges, such as current-carrying wires, and it can exert a force on other charged particles or magnetic materials.

Theorem: Ampère’s Circuital Law

Ampère’s circuital law relates the magnetic field to the electric current that generates it. The law states that the line integral of the magnetic field around a closed loop is proportional to the electric current passing through the loop:

$\oint\mathbf{H}\cdot d\mathbf{l} = \mathbf{J}$

where $\mathbf{J}$ is the current density vector.

Physics Formula: Magnetic Field of a Long Straight Wire

For a long straight wire carrying a current $I$, the magnetic field at a distance $r$ from the wire is given by:

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

where $\mu_0$ is the magnetic permeability of free space, with a value of $4\pi\times 10^{-7}$ H/m.

Physics Example: Magnetic Field of a Solenoid

Consider a solenoid with $N$ turns and a cross-sectional area $A$. If a current $I$ is passed through the solenoid, it generates a magnetic field $\mathbf{B} = \mu_0nI$, where $n = N/L$ is the number of turns per unit length.

Physics Numerical Problem: Magnetic Field of a Long Straight Wire

A long straight wire carries a current $I = 10$ A. What is the magnetic field at a distance $r = 2$ cm from the wire?

Solution:
Using Ampère’s law, we have:
$\oint\mathbf{B}\cdot d\mathbf{l} = \mu_0I$
Simplifying, we get:
$B(2\pi r) = \mu_0I$
Solving for $B$, we get:
$B = \mu_0\frac{I}{2\pi r} = 4\pi\times 10^{-7}\frac{10}{2\pi\times 0.02} = 0.1$ T

Magnetic Induction ($\mathbf{B}$)

Magnetic induction, also known as magnetic flux density, is denoted as $\mathbf{B}$ and is measured in Tesla (T) in the SI system. It is a measure of the magnetic flux through a given area and is related to the magnetic field through the relationship $\mathbf{B} = \mu\mathbf{H}$, where $\mu$ is the magnetic permeability of the medium.

Theorem: Faraday’s Law of Induction and Lenz’s Law

Faraday’s law of induction states that the induced electromotive force (EMF) in a conductor is proportional to the rate of change of the magnetic flux through the conductor:

$\varepsilon = -N\frac{d\Phi}{dt}$

where $\varepsilon$ is the induced EMF, $N$ is the number of turns in the coil, and $\Phi$ is the magnetic flux through the coil.

Lenz’s law states that the direction of the induced current is such that it opposes the change in the magnetic field that induced it.

Physics Formula: Relationship between Magnetic Field and Magnetic Induction

The relationship between the magnetic field $\mathbf{H}$ and the magnetic induction $\mathbf{B}$ is given by:

$\mathbf{B} = \mu\mathbf{H}$

where $\mu$ is the magnetic permeability of the medium.

Physics Example: Induced EMF in a Changing Magnetic Field

Consider a solenoid with $N$ turns and cross-sectional area $A$. If a current $I$ is passed through the solenoid, it generates a magnetic field $\mathbf{B} = \mu_0nI$, where $n = N/L$ is the number of turns per unit length. If the current is changing, then there is a changing magnetic flux through the solenoid, which induces an EMF according to Faraday’s law:

$\varepsilon = -N\frac{d\Phi}{dt} = -NA\frac{dB}{dt} = -\mu_0nAN\frac{dI}{dt}$

Physics Numerical Problem: Induced EMF in a Changing Magnetic Field

A circular coil of radius $R = 10$ cm has $N = 100$ turns and is placed in a uniform magnetic field $\mathbf{B} = 0.5$ T directed perpendicular to the plane of the coil. If the magnetic field is decreasing at a rate of $d\mathbf{B}/dt = -0.1$ T/s, what is the induced EMF in the coil?

Solution:
Using Faraday’s law, we have:
$\varepsilon = -N\frac{d\Phi}{dt} = -N\pi R^2\frac{dB}{dt} = -100\pi(0.1)^2(-0.1) = 0.314$ V

Physics Numerical Problem: Induced EMF in a Moving Coil

A rectangular coil of width $w = 5$ cm and length $l = 10$ cm is moving with a velocity $\mathbf{v} = 2$ m/s in a uniform magnetic field $\mathbf{B} = 0.5$ T directed perpendicular to the plane of the coil. What is the induced EMF in the coil?

Solution:
The magnetic flux through the coil is given by:
$\Phi = \mathbf{B}\cdot\mathbf{A} = \mathbf{B}lw$
The rate of change of the magnetic flux is:
$\frac{d\Phi}{dt} = \mathbf{B}l\frac{dw}{dt} = \mathbf{B}lv$
Using Faraday’s law, we have:
$\varepsilon = -N\frac{d\Phi}{dt} = -N\mathbf{B}lv = -N(0.5)(0.1)(2) = -0.01$ V

Figures, Data Points, Values, and Measurements

• Magnetic field strength: $\mathbf{H}$ is measured in A/m (ampere per meter) in the SI system.
• Magnetic induction: $\mathbf{B}$ is measured in Tesla (T) in the SI system.
• Magnetic permeability: $\mu$ is measured in Henry per meter (H/m) in the SI system.
• Magnetic flux: $\Phi$ is measured in Weber (Wb) in the SI system.
• Electromotive force: $\varepsilon$ is measured in Volts (V) in the SI system.
• Current: $I$ is measured in Amperes (A) in the SI system.
• Charge: $q$ is measured in Coulombs (C) in the SI system.
• Velocity: $\mathbf{v}$ is measured in meters per second (m/s) in the SI system.
• Electric field: $\mathbf{E}$ is measured in Volts per meter (V/m) in the SI system.
• Force: $\mathbf{F}$ is measured in Newtons (N) in the SI system.

References

1. Magnetic flux density – Encyclopedia Magnetica. (2023-09-29). Retrieved from https://www.e-magnetica.pl/doku.php/magnetic_flux_density
2. EC-5 MAGNETIC INDUCTION. (n.d.). Retrieved from https://www.physics.wisc.edu/instructional/phys104/EC5/EC-5.pdf
3. Tutorial: a beginner’s guide to interpreting magnetic susceptibility … (2022-04-19). Retrieved from https://www.nature.com/articles/s42005-022-00853-y
4. Trying to understand the difference between Magnetic induction field (B) and Magnetic Field (H). (2022-01-03). Retrieved from https://www.reddit.com/r/AskPhysics/comments/ruznt6/trying_to_understand_the_difference_between/
5. Measuring g using magnetic induction – IOPscience. (2023-02-21). Retrieved from https://iopscience.iop.org/article/10.1088/1361-6552/acb033