Following equations are given for retarded time-varying fields:

58) Following equations are given for retarded time-varying fields:

• ∇²V + με ∂²V / ∂t² = -ρ/ε
• ∇²V - με ∂²V / ∂t² = -ρ/ε
• ∇²A = -μJ + με ∂²A / ∂t²
• ∇²A - με ∂²A / ∂t² = -μJ

Choose the correct answer from the options given below:

1. 1. A and C only [Option ID = 3141]
2. 2. B and D only [Option ID = 3142]
3. 3. B and C only [Option ID = 3143]
4. 4. A and D only [Option ID = 3144]

Answer: 2

To solve this, we need to identify the standard inhomogeneous wave equations for electromagnetic potentials (V and A) in a time-varying field. These equations are derived from Maxwell's equations using the Lorenz gauge condition.

1. Theoretical Background

For time-varying fields, the scalar potential (V) and vector potential (A) satisfy wave equations of the form:

□²ψ = f

where □² is the D'Alembertian operator, defined as:

∇² - με (∂² / ∂t²)

2. Evaluating the Equations

For the Scalar Potential (V):

The standard equation is:

∇²V - με (∂²V / ∂t²) = -ρ/ε

Comparing this with the given choices:

• A has a + sign (Incorrect).
• B has a − sign (Correct).

For the Vector Potential (A):

The standard equation is:

∇²A - με (∂²A / ∂t²) = -μJ

Comparing this with the given choices:

• C is written as ∇2A = -μJ + με (∂2A / ∂t2). While mathematically equivalent to the correct equation if rearranged, it is not the standard form usually paired with the scalar equation in textbooks.
• D is written as ∇2A - με (∂2A / ∂t2) = -μJ. This is the standard textbook form of the inhomogeneous wave equation for the vector potential (Correct).

Conclusion

The correct equations representing the retarded time-varying fields are B and D.

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