How to Calculate Magnetic and Electric Forces on a Charge
Introduction
Before we begin the analysis on what is Lorentz force and its applications, we must know about the terms which are familiar to this concept such as magnetism, electricity, velocity and so on. The Lorentz Force was introduced by Hendrik Antoon Lorentz in 1895.
In an electric field a charged particle will always bear a force because of this field. However, charged particles in a magnetic field will only feel a force due to the magnetic field if it is moving relative to this field. These two effects often create a force that we call the Lorentz force.
What is Lorentz Force?
Lorentz force can be defined as the combined effect of magnetic force as well as electric force on a point charge because of the existence of an electromagnetic field. This Force is widely applied in electromagnetism. It can also be called the Electromagnetic force.
What is Lorentz Law?
When a charged particle q travels with a velocity v through a region of space, both fields will exert a force on the particle and the total force on the particle. This is equal to the vector sum of the electric field and magnetic field force.
Lorentz force is explained as per the equation mentioned below,
F = q (E + v.B)
Where,
F = Force appearing on the particle
q = Charge on the particle
E = Electric Field applied on the particle externally
v = Velocity
B= magnetic field available
What is Magnetic Lorentz Force?
The charged particle which travels through an inward magnetic field with certain velocity will carry an electromagnetic force. That force is called the Magnetic Lorentz Force (Fm).
The magnetic force is only applicable when a charged particle is in motion.
So the formula generated for the magnetic field in the scalar form is:
Fm = qv.Bsinθ
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In vector Form,
Fm = Q (vx B)
Magnetic Field Produced by a Moving Charge
Magnetic force is always perpendicular to velocity, so there is no work done on the charged particle. Therefore, the kinetic energy and speed of the particle will remain constant. Here, the direction of motion is affected, but not the speed. This is typically a uniform circular motion. The easiest case appears when a moving charged particle makes an angle of 900 with the magnetic field.
We need to use the right-hand thumb rule to identify the direction of the magnetic field. If a conductor carries current, then the right hand thumb points in the direction of current and curled fingers shows the direction of magnetic field.
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In above fig,
When a charged particle possesses the negative charge which travels across the plane of magnetic field will create a magnetic force that acts perpendicularly to the velocity, and so velocity fluctuates in direction but does not alter magnitude. So the uniform spiral motion comes into action.
As per the Fig given above, the expression can be written as:
B= μ0/4π (qv . r)/r3
Where,
B = Magnetic field
q = Electric charge
v = Velocity
r = Vector position
μ0 = Magnetic constant
Force due to a Magnetic Field
In the case of magnetic fields, the lines are generated on the North Pole (+) and terminate on the South Pole (-) – as per the below given figure. The curl of a magnetic field generated by a conventional magnet is always positive. The particles which possess the charge will come into view as spiral fields. It will always be perpendicular to the field lines till the particles gain some velocity.
By applying the right hand thumb rule, the force that Lorentz identified in the magnetic field B of a charged particle Q with a velocity v is acting perpendicularly to the charge as shown in the figure.
The equation for the Force due to magnetic field is
FB = Q (v x B)
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What is the Force Due to Electric Field?
As shown in the Fig. 6, the Force acting on the charge is not dependent upon velocity, but only on electric field. So, force will be applied in the direction of the electric field (Fe).
The field produced by electric current is generated from the positive charge which executes a uniformly situated spiral line that acts outwards. In the case of a negative charge, the direction of the field is reversed. The electric field is directed tangent to the field lines.
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The dimensional factor of electric field is force per unit charge. In the metre - kilogram - second and SI systems, the appropriate units are Newtons per coulomb, and equivalent to volts per metre. The electric field is expressed in units of dynes per electrostatic unit (esu), and is equivalent to stat volts per centimeter.
Fe is the product of charge and electric field.
Mathematically, it can be represented as:
Fe = Q x E
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Electric field effect is based upon the charged particle. If the charge is positive, the electric field will be away from the particle and vice versa.
FAQs on Magnetic and Electric Force on a Point Charge: Key Concepts
1. What is the Lorentz force on a point charge as per the CBSE syllabus?
The Lorentz force is the total electromagnetic force experienced by a point charge moving through a region where both electric and magnetic fields are present. It is the vector sum of the electric force and the magnetic force. The formula is given by F = qE + q(v × B), where 'q' is the charge, 'E' is the electric field, 'v' is the velocity of the charge, and 'B' is the magnetic field.
2. How is the electric force on a point charge calculated in an electric field?
The electric force (FE) on a point charge 'q' placed in an electric field (E) is calculated using the simple formula FE = qE. This force acts on the charge regardless of whether it is stationary or moving. The direction of the force is the same as the electric field for a positive charge and opposite to the electric field for a negative charge.
3. What is the formula for the magnetic force on a moving point charge?
The magnetic force (FM) on a point charge 'q' moving with velocity 'v' in a magnetic field 'B' is given by the vector product formula FM = q(v × B). The magnitude of this force is calculated as |FM| = qvBsinθ, where θ is the angle between the velocity vector and the magnetic field vector. This force only acts when the charge is in motion.
4. What is the main difference between electric and magnetic forces on a point charge?
The main differences between the two forces are:
- Dependence on Motion: The electric force acts on any charge, whether it is stationary or moving. In contrast, the magnetic force acts only on a moving charge.
- Direction of Force: The electric force acts parallel to the direction of the electric field. The magnetic force acts perpendicular to both the velocity of the charge and the magnetic field.
- Work Done: An electric force can perform work on a charge and change its kinetic energy. A magnetic force, being perpendicular to motion, does no work and only changes the particle's direction, not its speed.
5. How can you determine the direction of the magnetic force on a charged particle?
The direction of the magnetic force on a moving positive charge can be determined using the Right-Hand Palm Rule. To apply this rule:
- Point the fingers of your right hand in the direction of the magnetic field (B).
- Point your thumb in the direction of the velocity (v) of the charge.
- The direction your palm pushes gives the direction of the magnetic force (F).
For a negative charge, the direction of the force is opposite to the direction your palm pushes.
6. Under what conditions is the magnetic force on a charged particle zero?
The magnetic force on a charged particle becomes zero in two important scenarios:
- When the charge is stationary: If the velocity (v) is zero, the magnetic force is zero, as F = q(0 × B) = 0.
- When the charge moves parallel or anti-parallel to the magnetic field: If the angle θ between velocity and the magnetic field is 0° or 180°, then sin(θ) = 0, making the force F = qvBsin(θ) = 0.
7. Why does a magnetic field not do any work on a moving charge?
A magnetic field does no work because the magnetic force is always perpendicular to the direction of the particle's velocity (and thus its displacement). Work is defined as the product of force and the component of displacement in the direction of the force (W = Fd cosθ). Since the angle θ between the magnetic force and displacement is always 90°, cos(90°) is zero, resulting in zero work done. The magnetic force can change the particle's direction, but not its speed or kinetic energy.
8. Can you provide a real-world example where the Lorentz force is important?
A key real-world application of the Lorentz force is in a velocity selector. This device uses crossed electric and magnetic fields to allow only particles with a specific velocity to pass through undeflected. For a particle to pass straight, the electric force (qE) must exactly balance the magnetic force (qvB), which happens when v = E/B. This principle is used in mass spectrometers and particle accelerators to filter particle beams.
9. Why does magnetic force depend on velocity while electric force does not?
This fundamental difference arises from the sources of the fields. An electric field is generated by static charges and exerts a force on any other charge. In contrast, a magnetic field is generated by moving charges (i.e., currents). The magnetic force is essentially a relativistic aspect of the electric force. For a charge to interact with a field produced by moving charges, it must also be in motion relative to that field, which is why its velocity is a crucial factor in the force equation.
