Calculating Magnetic Force

Magnetic force is a consequence of electromagnetic force and is caused due to the motion of charges. We have learned that a moving charge surrounds itself with a magnetic field. With this context, the magnetic force can be described as a force that arises due to interacting magnetic fields. Learn more about magnetic force in detail.

How To Find Magnetic Force?

The magnitude of the magnetic force depends on how much charge is in how much motion in each of the objects and how far apart they are.

Mathematically, we can write magnetic force as:

\begin{array}{l} F = q [E (r) + v \times B (r)] \end{array}

This force is termed as the Lorentz Force. It is the combination of the electric and magnetic force on a point charge due to electromagnetic fields. The interaction between the electric field and the magnetic field has the following features:

The magnetic force depends upon the charge of the particle, the velocity of the particle and the magnetic field in which it is placed. The direction of the magnetic force is opposite to that of a positive charge.
The magnitude of the force is calculated by the cross product of velocity and the magnetic field, given by q [ v × B ]. The resultant force is thus perpendicular to the direction of the velocity and the magnetic field, the direction of the magnetic field is predicted by the right-hand thumb rule.
In the case of static charges, the total magnetic force is zero.

Example

Calculating Magnetic Force

Magnetic force is the force exerted by a magnetic field on a moving charged particle or current-carrying conductor. The magnitude of the magnetic force depends on the charge, velocity, magnetic field strength, and angle between the velocity and magnetic field lines.

The general approach to calculating magnetic force involves:

Identifying the charge, velocity, magnetic field strength, and the angle between them.
Using the appropriate formula for calculating magnetic force.

Magnetic Force on a Moving Charge Formula

The formula for the magnetic force on a moving charged particle is:

\[ F = qvB \sin(\theta) \]

Where:

F is the magnetic force (in newtons, N).
q is the charge of the particle (in coulombs, C).
v is the velocity of the particle (in meters per second, m/s).
B is the magnetic field strength (in tesla, T).
\(\theta\) is the angle between the velocity and the magnetic field (in degrees or radians).

Example:

If a proton (charge \( 1.6 \times 10^{-19} \, \text{C} \)) is moving with a velocity of \( 5 \times 10^6 \, \text{m/s} \) in a magnetic field of strength \( 0.2 \, \text{T} \), and the angle between the velocity and the magnetic field is \( 90^\circ \), we can calculate the magnetic force as:

Step 1: Use the magnetic force formula: \( F = qvB \sin(\theta) \).
Step 2: Substitute the known values: \( F = (1.6 \times 10^{-19} \, \text{C}) (5 \times 10^6 \, \text{m/s}) (0.2 \, \text{T}) \sin(90^\circ) \).
Step 3: Calculate the result: \( F = 1.6 \times 10^{-13} \, \text{N} \).

Magnetic Force on a Current-Carrying Wire Formula

The formula for the magnetic force on a current-carrying wire is:

\[ F = I L B \sin(\theta) \]

Where:

F is the magnetic force (in newtons, N).
I is the current flowing through the wire (in amperes, A).
L is the length of the wire within the magnetic field (in meters, m).
B is the magnetic field strength (in tesla, T).
\(\theta\) is the angle between the wire and the magnetic field (in degrees or radians).

Example:

If a current of \( 3 \, \text{A} \) is flowing through a wire of length \( 0.5 \, \text{m} \) placed in a magnetic field of \( 0.4 \, \text{T} \), with the wire making an angle of \( 90^\circ \) with the magnetic field, the magnetic force can be calculated as:

Step 1: Use the formula for the magnetic force on a wire: \( F = I L B \sin(\theta) \).
Step 2: Substitute the known values: \( F = (3 \, \text{A}) (0.5 \, \text{m}) (0.4 \, \text{T}) \sin(90^\circ) \).
Step 3: Calculate the result: \( F = 0.6 \, \text{N} \).

Right-Hand Rule for Magnetic Force

The right-hand rule can be used to determine the direction of the magnetic force. For a positive charge, point your thumb in the direction of the velocity, your fingers in the direction of the magnetic field, and your palm will point in the direction of the magnetic force. For a negative charge, the force is in the opposite direction.

Real-life Applications of Magnetic Force

Magnetic forces are fundamental in many technological and industrial applications. Some real-life applications include:

Designing electric motors, where magnetic forces are used to generate motion.
Generating electricity in generators, where a wire moves through a magnetic field to produce electrical current.
Operating MRI machines, which use powerful magnetic fields to generate detailed images of the inside of the body.

Common Units for Magnetic Force

SI Unit: The unit for magnetic force is the newton (N), and for magnetic field strength, it is the tesla (T).

The calculation of magnetic force helps to explain many phenomena in electromagnetism and is essential in the design of various electrical and mechanical systems.

Common Operations with Magnetic Force

Solving for Unknown Variables: You can solve for any of the variables in the magnetic force equations if you have the other values. For example, to find the velocity of a charged particle, rearrange the formula to \( v = \frac{F}{qB \sin(\theta)} \).

Force Direction and Magnitude: In systems where the magnetic force is acting on charged particles or conductors, understanding both the magnitude and direction of the force is crucial for the operation of devices like motors and generators.

Calculating Magnetic Force Examples Table
Problem Type	Description	Steps to Solve	Example
Calculating Magnetic Force on a Moving Charge	Finding the magnetic force on a moving charged particle in a magnetic field.	Identify the charge \( q \) of the particle (in coulombs). Identify the velocity \( v \) of the particle (in meters per second). Identify the magnetic field strength \( B \) (in teslas). Identify the angle \( \theta \) between the velocity and magnetic field. Use the magnetic force formula: \( F = qvB \sin(\theta) \).	If the charge is \( 2 \, \text{C} \), the velocity is \( 5 \, \text{m/s} \), the magnetic field is \( 3 \, \text{T} \), and the angle is \( 90^\circ \), the magnetic force is \( F = (2)(5)(3) \sin(90^\circ) = 30 \, \text{N} \).
Calculating Force on a Current-Carrying Wire	Finding the force on a current-carrying wire in a magnetic field.	Identify the current \( I \) flowing through the wire (in amperes). Identify the length \( L \) of the wire (in meters). Identify the magnetic field strength \( B \) (in teslas). Identify the angle \( \theta \) between the wire and the magnetic field. Use the formula: \( F = ILB \sin(\theta) \).	If the current is \( 3 \, \text{A} \), the length of the wire is \( 2 \, \text{m} \), the magnetic field is \( 4 \, \text{T} \), and the angle is \( 90^\circ \), the force is \( F = (3)(2)(4) \sin(90^\circ) = 24 \, \text{N} \).
Calculating Force in a Solenoid	Finding the magnetic force in a solenoid given the number of turns, current, and magnetic field.	Identify the number of turns \( N \) in the solenoid. Identify the current \( I \) passing through the solenoid (in amperes). Identify the magnetic field strength \( B \) (in teslas). Use the formula for magnetic force: \( F = NIBL \), where \( L \) is the length of the solenoid.	If the solenoid has \( 100 \) turns, a current of \( 5 \, \text{A} \), and a length of \( 0.5 \, \text{m} \) in a magnetic field of \( 2 \, \text{T} \), the force is \( F = (100)(5)(2)(0.5) = 500 \, \text{N} \).
Calculating Force in a Charged Particle Beam	Determining the force acting on a beam of charged particles in a magnetic field.	Identify the charge \( q \) of the particles. Identify the velocity \( v \) of the particles. Identify the number of particles \( N \) in the beam. Use the formula: \( F = NqvB \sin(\theta) \).	If the charge is \( 3 \, \text{C} \), the velocity is \( 7 \, \text{m/s} \), the number of particles is \( 50 \), and the magnetic field is \( 1.5 \, \text{T} \), the force is \( F = (50)(3)(7)(1.5) \sin(90^\circ) = 1575 \, \text{N} \).