Lorentz Force Law

Introduction

The Lorentz force is the total force exerted on a charged particle that’s moving through a region containing both an electric field and a magnetic field.

This force is fundamental to electromagnetism and is responsible for many phenomena, including

  • how electric motors work and
  • how charged particles are deflected in particle accelerators.

In this post, we’ll describe this two part equation, and we’ll start with a review of some vector cross product rules and conventions.

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Vector Cross Product Math Rules

Let’s denote c̅ = 

  • = vector a
  • = vector b
  • = cross product of vectors a̅ and b̅

From vector math , we know that

Vector Cross Product RHR Method 1

  • Curl the fingers of your right hand from the direction of 
  •  to the direction of  (through the smaller angle between them).
  • Your thumb will point in the direction of .
Picture: Vector Cross Product Right Hand Rule Method 1 (Curl Method)

Vector Cross Product RHR Method 2

  • Index finger towards 
  • , middle finger towards
  • thumb points in direction of the cross product
Picture: Vector Cross Product Right Hand Rule Method 2 (Index, Middle, Thumb)

Vector Cross Product RHR Method 3

  • Flat open hand, thumb towards 
  •  and other fingers toward
  • vector comes out of open palm side of hand.
Picture: Vector Cross Product Right Hand Rule Method 3 (Open Hand)

The magnitude of c̅ is given by the formula:

| ; Magnitude of 

where 

  • is the angle between  and  ().
Picture: 3D Coordinates Vector Cross Product :

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Force on a Charge in an Electric Field

We can manipulate Coulomb’s Law to derive some useful expressions: 

e = kqtqs/r; Coulomb’s Law
E̅ = F̅e/qt = kqs/r2 ; Electric Field 
e = qtE̅ ; Electric Force as a function of Charge and Electric Field

where,

  • F= electric force  (a vector)
  • E = electric field = electric field strength = Force/Charge (a vector)
  • qs = source charge (produces the electric field)
  • qt = test charge (entering the electric field of qs)
  • k = coulomb’s constant
  • SI units for E: Newtons/Coulomb = N/C
  • Other SI units for E = Nm/Cm = Volts/meter = V/m
  • N = newton
  • C = coulomb
  • V = voltage
  • M = meter

The schematic below shows a force F acting on a test charge qin an electric field E.

  • Remember that the F and E are vectors while q is a scalar.  
Picture: Charge q in Electric Field E

Characteristics of a charge in an Electric Field

  • A charge placed in an electric field will experience an electric force .
  • The direction of the force on a positive charge () is in the same direction as the electric field .
  • The direction of the force on a negative charge () is in the opposite direction to the electric field .
  • The magnitude of the force is directly proportional to the magnitude of the charge and the strength of the electric field.

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Force on a Moving Charge in a Magnetic Field

Now let’s put a positive test charge q with a velocity v into a magnetic field B.

The force equation in vector notation is:

B = qt(v̅ x B̅) ; Force on a Test Charge in a Magnetic Field (vector form)

The magnitude form of the equation is

FB = qtvBsinθ ; Magnitude of Force on a Test Charge in a Magnetic Field 

Where

  • FB = Force on charge q by magnetic field B
  • q= charge
  • vBsinθ = magnitude of (v̅ x B̅)
  • v = velocity of charge
  • B = Magnetic Field
  • θ = Angle between velocity and direction of B

Let’s rearrange the magnitude equation in terms of the magnetic field B:

B = FB/qtvsinθ

Let

  • q = 1 Coulomb = 1C
  • v = 1 meter/s = m/s
  • θ = 90 degrees

Then the equation becomes B = 1 FNewtons/(Coulomb-meters) = 1 FNs/Cm = 1 Tesla

  • 1 Ns/Cm = 1 Tesla

So we can say that the magnetic field B is the force per 1 unit charge 

where  

  • the charge is moving at 1 m/s  and 
  • the charge is perpendicular to the magnetic field. 

For example, B = 100 means 1 Coulomb of charge moving perpendicular to B at 1 m/s would experience a force of 100 N. 

Picture: Force on a Moving Charge in a Magnetic Field

Some key characteristics of the Magnetic Force are:

  • The magnetic force depends on the velocity of the charge.
    • If v is 0, F is zero
  • The magnetic force depends on the direction of motion.
    • Since sin 90 = 1 is maximum, the maximum force occurs when the speed is perpendicular to the direction of B. 
  • The magnetic force direction is always perpendicular to the magnetic field B
    • The electric force on the other hand is always parallel to the electric field E direction 
Application of the Cross Product Right Hand Rules

Because  B = qt(v̅ x B̅) and qis a scalar,  you can use any of the three Right Hand Rules to determine the direction of the Force component

If the charge is negative, then the direction of F will flip.   

Picture: Vector Cross Product Right Hand Rules Apply to the Magnetic Force on a Charge Equation 

 

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Synthesis of the Force on a Charge Expressions

In the 1890s, Hendrik Lorentz, a Dutch Physicist, was the first to combine these previously known electric and magnetic forces into a single, comprehensive law.

  • The electric force component was known from Coulomb’s law, and
  • the magnetic force component was formulated by Oliver Heaviside.

The Lorentz Force is the total electromagnetic force experienced by a charged particle moving through a region containing both an electric field (E) and a magnetic field (B). 

The Lorentz Force Equation (Law): 
 = + F̅B
 = qtE̅ + qt(v̅ x B̅)
or
 = qt (E̅ + (v̅ x B̅))

where

  • F̅ = Total Force. The Lorentz Force
  • B = Force on charge q by magnetic field B
  • e = Force on charge q by electric field E
  • qt= charge
  • v = velocity of charge
  • E̅ = electric field
  • B̅ = Magnetic Field
  • θ = Angle between velocity and direction of B
Electric Force Component (qtE̅):
  • represents the force exerted by the electric field E on the charge q.
  • The force is parallel to the electric field for a positive charge (q>0) and flipped 1809 degrees  for a negative charge (q<0).
  • This force acts on the charge whether it is moving or stationary.
Magnetic Force Component (qt(v̅ x B̅)):
  • represents the force exerted by the magnetic field B on a moving charge q.
  • It is a vector cross product, which means the magnetic force is always perpendicular to both the velocity v and the magnetic field B.
  • This force only acts on a charge if
    • it is moving and
    • if its velocity has a component perpendicular to the magnetic field.
  • The direction is determined by the right-hand rule of which there are three methods.

View these excellent videos for more information:

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