The relativistic equation of motion of a charged particle moving in an electromagnetic field can be written in covariant form using four-vectors and the electromagnetic field tensor. Which of the following equations correctly represents this relation?

Explanation: 1. Conceptual Framework To write a physical law in a covariant form (independent of the choice of reference frame), all quantities in the equation must change properly under Lorentz transformations. This requires expressing the mechanics using tensors and four-vectors with respect to proper time ( ) rather than coordinate time ( ). Four-momentum ( ): Represents the relativistic energy and momentum vector. Four-velocity ( or ): The rate of change of position four-vector with respect to proper time ( ). Electromagnetic Field Tensor ( ): A rank-2 antisymmetric tensor containing both electric and magnetic field components. Proper Time ( ): An invariant scalar denoting time measured in the particle's rest frame. 2. Analysis of the Relativistic Lorentz Force The classical Lorentz force equation

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Explanation

1. Conceptual Framework

To write a physical law in a covariant form (independent of the choice of reference frame), all quantities in the equation must change properly under Lorentz transformations. This requires expressing the mechanics using tensors and four-vectors with respect to proper time ( $τ$ ) rather than coordinate time ( $t$ ).

Four-momentum ( $p^{μ}$ ): Represents the relativistic energy and momentum vector.
Four-velocity ( $U^{μ}$ or $U_{ν}$ ): The rate of change of position four-vector with respect to proper time ( $τ$ ).
Electromagnetic Field Tensor ( $F^{μν}$ ): A rank-2 antisymmetric tensor containing both electric and magnetic field components.
Proper Time ( $τ$ ): An invariant scalar denoting time measured in the particle's rest frame.

2. Analysis of the Relativistic Lorentz Force

The classical Lorentz force equation is given by:

$F = q (E + v \times B)$

In Minkowski space, this expression generalizes to a relation connecting the Minkowski four-force ( $K^{μ}$ ), the electromagnetic field tensor ( $F^{μν}$ ), and the four-velocity.

The four-force is defined as the derivative of four-momentum with respect to proper time $τ$ :

$K^{μ} = \frac{d p ^{μ}}{d τ}$

The interaction with the electromagnetic field tensor requires a contracted index matching to yield a clean four-vector result:

$K^{μ} = q F^{μν} U_{ν}$

3. Evaluating the Indices and Options

Let's break down why option (a) is structurally and physically valid:

Time Parameter: It must use proper time ( $d τ$ ) instead of coordinate time ( $d t$ ) to maintain full manifest covariance. This immediately eliminates choices (b) and (d).
Index Balance: The left side has a free contravariant index $μ$ . Therefore, the right side must also feature a free contravariant index $μ$ .
In the term $F^{μν} U_{ν}$ , the index $ν$ is repeated (once as a superscript in $F$ and once as a subscript in $U$ ). This means $ν$ is a dummy index summed over all four dimensions ( $0, 1, 2, 3$ ) following the Einstein summation convention.
This leaves $μ$ as the lone free index on both sides, ensuring consistent tensor transformation balance. In choice (c), the indices are improperly matched.

Thus, the exact tensor formulation matches option (a).

Correct Answer

The correct option is (a) $\frac{d p ^{μ}}{d τ} = q F^{μν} U_{ν}$ .

Explanation

1. Conceptual Framework

Four-momentum ( $p^{μ}$ ): Represents the relativistic energy and momentum vector.
Four-velocity ( $U^{μ}$ or $U_{ν}$ ): The rate of change of position four-vector with respect to proper time ( $τ$ ).
Electromagnetic Field Tensor ( $F^{μν}$ ): A rank-2 antisymmetric tensor containing both electric and magnetic field components.
Proper Time ( $τ$ ): An invariant scalar denoting time measured in the particle's rest frame.

2. Analysis of the Relativistic Lorentz Force

The classical Lorentz force equation is given by:

$F = q (E + v \times B)$

In Minkowski space, this expression generalizes to a relation connecting the Minkowski four-force ( $K^{μ}$ ), the electromagnetic field tensor ( $F^{μν}$ ), and the four-velocity.

The four-force is defined as the derivative of four-momentum with respect to proper time $τ$ :

$K^{μ} = \frac{d p ^{μ}}{d τ}$

The interaction with the electromagnetic field tensor requires a contracted index matching to yield a clean four-vector result:

$K^{μ} = q F^{μν} U_{ν}$

3. Evaluating the Indices and Options

Let's break down why option (a) is structurally and physically valid:

Time Parameter: It must use proper time ( $d τ$ ) instead of coordinate time ( $d t$ ) to maintain full manifest covariance. This immediately eliminates choices (b) and (d).
Index Balance: The left side has a free contravariant index $μ$ . Therefore, the right side must also feature a free contravariant index $μ$ .
In the term $F^{μν} U_{ν}$ , the index $ν$ is repeated (once as a superscript in $F$ and once as a subscript in $U$ ). This means $ν$ is a dummy index summed over all four dimensions ( $0, 1, 2, 3$ ) following the Einstein summation convention.
This leaves $μ$ as the lone free index on both sides, ensuring consistent tensor transformation balance. In choice (c), the indices are improperly matched.

Thus, the exact tensor formulation matches option (a).

Correct Answer

The correct option is (a) $\frac{d p ^{μ}}{d τ} = q F^{μν} U_{ν}$ .

AMU 2026 — science PYQ

Choose the correct answer:

Explanation

1. Conceptual Framework

2. Analysis of the Relativistic Lorentz Force

3. Evaluating the Indices and Options

Correct Answer

Explanation

1. Conceptual Framework

2. Analysis of the Relativistic Lorentz Force

3. Evaluating the Indices and Options

Correct Answer