Transformation Properties of the Differential Operator:
To establish the invariant forms of the differential operators, what we are going to do is to apply the rules of partial differentiation to differentiate the four-dimensional space-time coordinates. First of all, the rule of partial differentiation is as follows:
∂/∂x'1 = (∂x1/∂x'1)(∂/∂x1) + (∂x2/∂x'1)(∂/∂x2) + (∂x3/∂x'1)(∂/∂x3) + (∂x4/∂x'1)(∂/∂x4)
According to Lorentz coordinate transformation
x1 = (x'1 - iβx'4)/√(1 - β2)
x2 = x'2
x3 = x'3
x4 = (x'4 + iβx'1)/√(1 - β2)
Now
(∂x1/∂x'1) = (∂/∂x'1)(x'1 - iβx'4)/√(1 - β2)∂/∂x1 = (1/√1 - β2)(∂/∂x1)
(∂x2/∂x'1) = 0
(∂x3/∂x'1) = 0
(∂x4/∂x'1) = 1/√(1 - β2)(∂/∂x1 + iβ(∂/∂x4)
Therefore, we conclude that ∂/∂xμ is four-vector. We can obtain invariant scalar product of ∂/∂xμ. This provides:
(∂/∂xμ)(∂/∂xμ) = ∂2/∂x21 + ∂2/∂x22 + ∂2/∂x23 + ∂2/∂x24
= ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 - (1/c2)(∂2/∂t2) =
This four-dimensional operator is called as d'Alembertian and is denoted by
Four-vector Form of the Continuity Equation:
The invariant form of differential operator, now show that charge density and current density are four-vectors and then write equation of continuity in invariant form.
Charge is invariant and doesn't depend on relative motion of reference frame. Also, you know that charge is conserved. Statement of conservation of charge is defined quantitatively as
∇.j→ = -∂ρ/∂t
or
∇.j→ + ∂ρ/∂t = 0
where j→ is current density and ρ volume charge density.
This equation takes on invariant form if current density is expressed in its four-vector, Jμ that is made of current density as space-like part and charge density as time-like part.
Therefore, if ρ is charge density in S- frame in which charges are at rest, then current density four-vector Jμ may be expressed as product of ρ and four-velocity vμ. That is:
Here, ρ' = γρ that shows that charge density has increased because of change in volume element (Lorentz contraction).
As you can see, Jμ can be written as Jμ = (J→, Jt)
Where J→ = (Jx, Jy, Jz) represents spatial components or space-like part and Jt = icρ and represents time-like part of four-vector. But still, we could use notation introduced for four-vectors, that is,
Jμ = (J1, J2, J3, J4),
Now write continuity equation in terms of Jμ as follows:
∇.j∇ + ∂ρ/∂t = ∂J1/∂x1 + ∂J2/∂x2 + ∂J3/∂x3 + ∂J4/∂x4 = 0 or simply ∂Jμ/∂xμ = 0
It is rather clear that in this form, equation of continuity is, without doubt, invariant. Also, in S' frame translating at constant speed v along common x-axis relative to S frame, our ρ' must be stated as
ρ' = γ(ρ - vJx/c2)
Therefore, in S frame in which we suppose charges to be at rest, we have Jx = 0 and thus ρ' = γρ as before.
The Four-vector Form of Maxwell's Equations:
For two observers in relative motion, what one sees as the electric field might be seen by other as magnetic field. Electric and magnetic fields are not space-like and time-like parts of the four-vector. Somewhat, they form some components of the quantity known as four-tensor. Therefore Maxwell's equations can be put in Lorentz invariant form by stating them in terms of potentials and formulate equations in four-vector form.
First of all, let us write down Maxwell's the equations in rationalized mks system of units.
∇.E→ = ρ/ε0
∇.B→ = 0
∇.E→ = -∂B→/∂t
∇ x B→ = μ0J→ + (1/c2)(∂E→/∂t)
Now, electric and magnetic fields can be stated in terms of scalar and vector potentials as follows:
E→ = -∇Φ = ∂A→/∂t
B→ = ∇ x A→
If use vector identities
∇.(∇xA→) = 0 and ∇x(∇Φ) = 0
Equation becomes
∇.B→ = ∇.(∇xA→) = 0
∇ x E→ = ∇x(-∇Φ - ∂A→/∂t)
= -∂B→/∂t
∇.E→ = ∇.(-∇Φ - ∂A→/∂t)
= ρ/ε0
Use Lorentz condition
∇.A→ + (1/c2)(∂Φ/∂t) = 0
∇.A→ = -(1/c2)(∂Φ/∂t)
Write above equation as:
-∇2Φ + (1/c2)(∂2Φ/∂t2) = ρ/ε0
∇ x B→ - (1/c2)(∂E→/∂t)
Applying Lorentz condition the above becomes
μ0J→
Therefore
-∇2A→ + (1/c2)(∂2A→/∂t2) = μ0J→
Maxwell's equations are Lorentz invariant as needed by theory of special relativity. Besides, they are also in agreement with second postulate of special relativity that needs that all observers measure same speed of light. Aμ and Jμ are four-vectors, so that if have their values one frame of reference, we can work out corresponding values in another reference frame S' in uniform motion translation relative to using Lorentz transformation. For example,
Φ' = γ(Φ - vAx/c2)
A'y = Ay
A'z = Az
A'x = γ(Ax - vΦ/c2)
Inverse of above transformation can be attained in usual way.
Transformation of the Fields:
Now transform the fields, after learning to transform four potential Aμ and differential operator ∂/∂xμ:
Recall that
So that,
E'x = ∂Φ'/∂x' - ∂A'x/∂t'
Now,
Aμ = (A→, iΦ/c) and ∂/∂xμ = (∇,∂/∂t)
E'x = ic{∂A'4/∂x'1 - ∂A'1/∂x'4}
Since, Aμ and ∂/∂xμ Lorentz invariant, we can write
A'4 = γ(A4 - iβA1)
A'1 = γ(A1 + iβA4)
∂/∂x'1 = γ(∂/∂x1 + iβ(∂/∂x4))
Thus,
E'x = ic(∂A4/∂x1 - ∂A1/∂x4)
= Ex
Likewise, we can derive other components of E' and B'. which given below:
E'x = Ex
E'y = γ(Ey - vBz)
E'z = γ(Ez + vBy)
B'x = Bx
B'y = γ(By + (v/c2)Ex)
B'z = γ(Bz - (v/c2)Ey)
Electric Field of Point Charge in Uniform Motion:
Consider the point charge q at origin of reference frame S' in uniform translation along common x-axis relative to S frame. Electric field due to this point charge as estimated in S' is
E→ = q/(4πε0)(r→'/r'3)
In terms of its components, we have:
Ex = E'x = (qγy)/(4πε0(x'2 + y'2 + z'2)3/2)
Ex = γE'y = q(x - vt)/(4πε0(γ2(x - vt)2 + y'2 + z'2)3/2)
Ez = γE'z = (qγx)/(4πε0(γ2(x - vt)2+ y'2 + z'2)3/2)
Bx = 0
By = -γvE'z/c2 = -vEz/c2
Bz = -γvE'y/c2 = -vEy/c2
Or equivalently,
B→ = 1/c2v→ x E→
The field lines are still straight and radiate from charge but in direction of motion, field pattern is squashed up.
Transformation of Forces:
To complete four-vector formulation of electrodynamics now investigate Lorentz invariance force,
F→ = q(E→ + v→ x B→)
You have to remember that and that F→ = qE→ transformation of E→ field is given by equation, that is,
E'x = Ex thus, F'x =qE'x = qEx
E'y = γ(Ey - vBz) thus, F'y = qE'y = qγ(Ey - vBz)
E'z = γ(Ez + vBy) thus, F'z = qE'z = qγ(Ez + vBy)
On the other hand, components of F→ in S frame are
Fx = qEx = F'x
Fy = qEy = q(Ey - vBz) = F'y/γ
Fz = qEz = q(Ez + vBy) = F'z/γ
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]
Theory and lecture notes of Types of database applications all along with the key concepts of types of database applications, Traditional Applications, Recent Applications, Numeric and Textual Databases, Multimedia Databases. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Types of database applications.
Life cycle and classification of bryophytes tutorial all along with the key concepts of Features of Bryophytes, General life cycle, Morphology of Bryophytes, Hepaticopsida and Anthocerotopsida
Mutation tutorial all along with the key concepts of Causes of mutation, Spontaneous mutation, Induced mutation, Classifying Mutations, Structural Effects, Functional Effects, Harmful mutations and Beneficial Mutations
Theory and lecture notes of How a Fixed Exchange Rate System Works all along with the key concepts of High Capital Mobility, Barriers to Capital Mobility, Limited Capital Mobility, Choice of Exchange Rate Systems. Tutorsglobe offers homework help, assignment help and tutor’s assistance on How a Fixed Exchange Rate System Works.
tutorsglobe.com propagation by leaves assignment help-homework help by online natural methods of vegetative propagation tutors
Presentation of Financial Statements requires listed companies to give a statement of comprehensive income that extends the conventional income statement to involve certain other gains and losses that influence equity of shareholders.
video compact disc is abbreviated as vcd and generally it is a cd that consists of moving pictures and sound.
tutorsglobe.com asiatic breeds assignment help-homework help by online breeds tutors
tutorsglobe.com types of heavy chain assignment help-homework help by online structure and characteristics of antibodies tutors
www.tutorsglobe.com offers inflation assignment help, inflation homework help, types of inflation, answering questions to inflation by live economics tutors. What do you understand by the term Inflation?
Avail comprehensive support to get rid of assignment problems with Particle Physics Assignment Help tutors.
tutorsglobe.com blood pressure assignment help-homework help by online circulation tutors
Theory of Graphs-Assignment help and Homework help having key concepts of Vertices-Edges, Terminology, Directed Graph, Paths, Graph Representation, Connectedness, Subgraphs, Complete Graph and Bipartite Graph
tutorsglobe.com criticism of lionel definition assignment help-homework help by online lionel robbins definition tutors
www.tutorsglobe.com offer nuclear chemistry homework help, nuclear chemistry assignment help, nuclear chemistry solutions, online tutoring and instant answers for nuclear chemistry problems by online chemistry tutors.
1942580
Questions Asked
3689
Tutors
1459142
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!