Okay, let's try to figure out this problem. So, the question is about finding the scalar potential V and vector potential A at a point r due to a moving point charge q. The charge is moving along a trajectory s(t), and we need to consider the potentials at time t, where t is greater than the retarded time tr. The options are given, and we need to pick the correct one.

First, I remember that when dealing with potentials due to a moving charge, especially when the charge is accelerating, we have to consider both the electric and magnetic contributions. In the case of a charge moving at a constant velocity, the potentials can be expressed using the retarded time. But here, since the charge is moving with a trajectory s(t), its velocity is changing, so the potentials will involve some integration over the past motion.

Wait, but the problem mentions that we should use the notation for the speed of light c, permittivity ε₀, and permeability μ₀. So, the potentials must involve these constants. The options include expressions with these constants, so that's reassuring.

The first thing I recall is that the scalar potential V for a moving charge is given by the integral of the charge density divided by the distance. Since the charge is a point charge, the scalar potential should be similar to the Coulomb potential, but with a time delay. The Coulomb potential is q/(4πε₀ r), so maybe V is the same here but with r replaced by the retarded distance? But the options have V as q/(4πε₀ r) in options A, B, and C. Option D has a different expression for V. Wait, but if the charge is moving, shouldn't the potential depend on the retarded distance? But in the problem statement, the potentials are given at time t, and the field is observed at that time. However, the potentials are expressed in terms of r, the position vector of the observation point, not the position at the retarded time. Hmm, that's confusing.

Wait, maybe I need to clarify: the potentials at a point r and time t due to a moving charge. If the charge is moving, then the potentials should be evaluated at the retarded time tr, not at the current time t. But in the options, the potentials are given in terms of r and t. So perhaps the potentials are expressed by integrating over the past trajectory?

Alternatively, maybe the problem is using the Lienard-Wiechert potentials. I think those are the potentials for a moving charge, which include the velocity and acceleration terms. Let me recall. The scalar potential is given by φ = (q / (4πε₀)) * (1 / (r - (r·v̂) / c²)), but that's when considering the Lorentz factor. Wait, maybe that's the Liénard-Wiechert potential. Let me think.

Wait, the Liénard-Wiechert potentials are given by:

φ = (q / (4πε₀)) * (1 / (r - (r·v̂) / c²))

and the vector potential A is:

A = (μ₀ q / 4π) * (v̂ × (r - (r·v̂) / c²)) / (r - (r·v̂) / c²)

But I might be mixing up the exact expressions. Alternatively, another form is:

φ = (q / (4πε₀)) * (1 / (r')) * (1 - (v²)/(c²))^{1/2}

where r' is the distance from the charge to the observation point, but evaluated at the retarded time. Wait, maybe not. Let's think again.

Alternatively, the Liénard-Wiechert potentials can be written as:

φ = (q / (4πε₀)) * (1 / |r - (1/c)(v × (r - vt)) × t))

Hmm, perhaps that's complicating things. Maybe the problem is simpler. The question mentions that the potentials should be expressed in terms of the vector d, which is the vector from the position where the field was generated at time tr to the observation point r. So, d = r - s(tr), where s(tr) is the position of the charge at the retarded time tr.

Wait, the problem says: "Let d be the vector from the point where the electromagnetic field was generated at some earlier time tr to the observation point r". So, d = r - s(tr). That's correct. Then, the velocity of the charge at the generating instant tr is v.

So, the potentials at time t (t > tr) would depend on the charge's position at the retarded time tr and its velocity at that time. Therefore, the potentials should involve the distance d, and the velocity v at tr.

But the options given don't all express the potentials in terms of d. Let's check the options:

Option A: V = q/(4πε₀ r), A = (v²/c²) V

Option B: V = q/(4πε₀ r), A = (v/c²) V

Option C: V = qc/(4πε₀ (d c - d·v)), A = μ₀ qc v/(4π (d c - d·v))

Option D: V = qc/(4πε₀ (d c + d·v)), A = μ qc v/(4π (d c + d·v))

Wait, but the standard Liénard-Wiechert expressions for the potentials involve terms like (1 - (v²)/(c²)) and terms involving (1 - (v·d)/c²) in the denominator. Let me try to recall the exact expressions.

The scalar potential φ is given by:

φ = (q / (4πε₀)) * (1 / |d|) * (1 - (v²)/(c²))^{1/2}

But wait, that's not exactly right. Alternatively, the scalar potential at a point r and time t is given by the charge divided by 4πε₀ times the reciprocal of the distance from the charge's position at the retarded time, multiplied by the Lorentz factor (1 - v²/c²)^(-1/2). Wait, I'm getting confused.

Alternatively, the Liénard-Wiechert potentials are:

φ = (q / (4πε₀)) * (γ (1 - (v·n)/c²))^{-1} / |d|

where n is the unit vector in the direction of d, and γ is the Lorentz factor 1/sqrt(1 - v²/c²). But substituting γ into this expression, we get:

φ = (q / (4πε₀)) * (1 / (|d| (1 - (v·n)/c²)))

But since n = d / |d|, this becomes:

φ = (q / (4πε₀)) * (1 / (|d| - (v·d)/c²))

Similarly, the vector potential A is given by:

A = (μ₀ q / (4π)) * (γ v / (|d| (1 - (v·n)/c²)))

But substituting γ, this becomes:

A = (μ₀ q / (4π)) * (v / (|d| (1 - (v·n)/c²)))

But let's check the options. The options involve expressions with d c ± (d·v). Let's see.

In option D, the denominator is (d c + d·v). Let's see. If we take the expression for φ from the Liénard-Wiechert formula:

φ = (q / (4πε₀)) * 1 / (|d| - (v·d)/c²)

But |d| is the magnitude of d, so |d| = sqrt(d·d). Let's see if we can manipulate this expression. Let's factor out |d| from the denominator:

|d| - (v·d)/c² = |d| [1 - (v·d)/(c² |d|)] = |d| [1 - (v/(c²))·(d/|d|)]

Which is |d| [1 - v·n / c²], where n is the unit vector in the direction of d. So, this matches the expression I had earlier.

But the options have denominators like (d c ± (d·v)). Let's compute:

If d c is the magnitude |d| multiplied by c, but that might not make sense. Wait, perhaps there's a confusion in notation here. Let's check the options again.

Option D: V = qc/(4πε₀ (d c + d·v)), A = μ qc v/(4π (d c + d·v))

Wait, the denominator here is (d c + d·v). Let's see. Let's suppose that |d| is involved here. If d·v is the dot product, then d c would be the magnitude of d multiplied by c. But in the denominator, it's written as d c + d·v. That seems odd. Wait, maybe the options are using d as the vector, but in the denominator, it's written as a scalar. For example, in option D, the denominator is (d c + d·v). But d is a vector, so this would be a vector plus a scalar. That doesn't make sense. Wait, maybe there's a typo in the options. Let me check again.

Looking back at the problem statement, the options are written as:

Option C: V = qc/(4πε₀ (d c - d·v)), A = μ₀ qc v/(4π (d c - d·v))

Option D: V = qc/(4πε₀ (d c + d·v)), A = μ qc v/(4π (d c + d·v))

Wait, but if d is a vector, then (d c ± d·v) would be a vector plus a scalar, which is not a valid denominator. So, that can't be right. So perhaps the options have a mistake in notation, and the denominators are scalar expressions. Alternatively, maybe d c refers to |d| * c, but the problem states that d is the vector from the charge's position at tr to the observation point r. So, d is a vector, but in the options, the denominators are written as scalar expressions. So, that suggests that perhaps the options are using |d| in the denominator, but it's written as d c ± d·v. Hmm, that's confusing.

Alternatively, maybe the options are using |d| as the magnitude, but written as d c ± (d·v). Let's see. Let's compute |d| c ± (d·v). If we take |d| c + (d·v), that would be c |d| + d·v. But let's see if that's equivalent to something else.

Wait, another approach: maybe the potentials can be written in terms of the retarded distance. Let's recall that for the scalar potential, the standard expression is:

V = (q / (4πε₀)) * (1 / (r' - (v·r')/c²))

where r' is the distance from the charge to the observation point at the retarded time. But since the charge is moving, r' is |d|. Wait, but d is the vector from the charge's position at tr to r. So, |d| is the distance r'. Therefore, maybe the scalar potential is:

V = (q / (4πε₀)) * (1 / |d|) * (1 / (1 - (v·d)/(c² |d|)))

But that's getting complicated. Let's see if any of the options match this. Option D's V is qc/(4πε₀ (d c + d·v)). Let's compute the denominator:

d c + d·v = |d| c + (d·v)

But (d·v) is the dot product, which is |d||v| cosθ, where θ is the angle between d and v. So, unless |d| is small compared to |v|, this term can't be neglected. But in the expression for V, we have 1 over (|d| c + d·v). Wait, but |d| c is a term with units of distance times speed, which doesn't make sense. Wait, maybe there's a confusion here. Let me check the dimensions.

The scalar potential V has units of volts (V). Let's check each option:

Option A: (q / (4πε₀ r)) has units of (C / (F/m * m)) = C/(C²/(N·m²) * m) )= N·m²/(C) * 1/m = N·m/C = J/C = V, which is correct. The A term is (v²/c²) V, which is (m²/s²)/(m²/s²) * V = V, so correct.

Option B: Same V as A, and A is (v/c²) V, which would have units (m/s)/(m²/s²) * V = s/(m) * V. But A should have units of tesla·m² or (V·s)/m. Wait, let's see. The vector potential A has units of (Wb/m²) * m² = Wb, but Wb is V·s. Wait, alternatively, A has units of (V·s)/m. Let's check:

v has units m/s, c² has units m²/s². So v/c² has units (m/s)/(m²/s²) = s/m. Then multiplying by V (which is in volts), we get (s/m) * V. But the correct units for A are (V·s)/m. So (s/m) * V is equivalent to (V·s)/m, which matches. So the units check out.

Option C: V is qc/(4πε₀ (d c - d·v)). Let's check units. qc has units C * C? Wait, no, q has units C, c has units m/s. So qc has units C·m/s. Then denominator is (d c - d·v). d is a vector (position), so d·v is scalar (m/s * m = m²/s). d c would be m * m/s = m²/s. So denominator has units m²/s. Numerator: qc has units C·m/s. So overall units of V are (C·m/s) / (m²/s) )= C/(m) = (C/(N·m)) * N·m/m = (V/m) * m = V. So units check out. The A term in option C is μ₀ qc v/(4π (d c - d·v)). μ₀ has units H/m = (V·s²/C²)·m^{-1}. Then numerator: μ₀ q c v has units (V·s²/C²)·m^{-1} * C * m/s * m/s = (V·s²/C²)·m^{-1} * C·m²/s² = V·m. Denominator: same as before, m²/s. So overall units are (V·m) / (m²/s) )= V·s/m, which matches the units of A. So units check out here.

Option D: V has units (C·m/s) / (m²/s) )= C/(m) = V/m, which is not correct for a scalar potential. Wait, no: denominator is (d c + d·v). d is a vector, so d c is m * m/s = m²/s, and d·v is m/s * m = m²/s. So denominator is m²/s + m²/s = m²/s. Numerator: qc has units C·m/s. So V has units (C·m/s) / (m²/s) )= C/m = V/m. Wait, scalar potential has units V, not V/m. So this can't be right. Therefore, option D is incorrect in terms of units. Similarly, option C's V has units correct as V/m? Wait, no. Wait, in option C, V is (q c)/(4π ε₀ (d c - d·v)). Let's check again. d c is m * m/s = m²/s. d·v is m/s * m = m²/s. So denominator is m²/s. Then numerator is q c: C * m/s. So overall, (C·m/s) / (m²/s) )= C/m = V/m. But scalar potential is in volts, not V/m. Therefore, option C's V has incorrect units. Similarly, option D's V also has incorrect units. Wait, but earlier when I checked option D, I thought it might have correct units, but now I'm seeing that V in option D has units V/m. So both options C and D have V with units V/m, which is incorrect. Therefore, options C and D are wrong.

Wait, but let's re-examine option D's V. Option D says V = qc/(4πε₀ (d c + d·v)). Let's compute the units again. qc has units C·m/s. Denominator: d c is m*(m/s) = m²/s. d·v is m/s * m = m²/s. So denominator is m²/s. So V has units (C·m/s) / (m²/s) )= C/m = V/m. Which is not correct. So option D's V has incorrect units. Option C's V is similar: (q c)/(4π ε₀ (d c - d·v)) → units C·m/s / (m²/s) )= C/m = V/m. So units are wrong. So options C and D are invalid. Therefore, the correct answer must be among A and B.

Now, let's check the units for A in options A and B.

Option A: A = (v²/c²) V. V has units V, v² has units m²/s², c² has units m²/s². So (v²/c²) has units (m²/s²)/(m²/s²) = 1. So A has units V, which is correct.

Option B: A = (v/c²) V. v/c² has units (m/s)/(m²/s²) )= s/m. Then multiplied by V (V), gives (s/m) * V. Which is (V·s)/m, which is correct for A's units. So both options A and B have correct units for V and A.

Now, let's think about the correct expressions for the scalar and vector potentials. Since the charge is moving, the potentials are retarded. The scalar potential should be proportional to q/(4πε₀ r'), where r' is the distance from the charge to the observation point at the retarded time. But the options use r, not r'. Wait, but the problem states that t > tr, so the potentials are evaluated at t, but the charge was at position s(tr) at time tr. So the distance r' is |r - s(tr)|, but in the options, the potentials are written in terms of r and t. So perhaps the options are using the instantaneous distance r = |r - s(t)|, but that would be the case if the charge were static. Wait, no. For a moving charge, the potentials must be evaluated at the retarded time tr, not the current time t. However, the problem seems to present the potentials in terms of r and t, which is confusing.

Wait, the problem says: "the potentials at time t, satisfying t > tr, and position r". So perhaps the potentials are evaluated at time t, but the charge was at position s(tr) at time tr. Therefore, the potentials depend on the position of the charge at the retarded time. Therefore, the potentials should involve the distance between r and s(tr), which is d. So, the potentials are expressed in terms of d and the velocity at tr.

But in the options, the potentials are written in terms of r and t. So perhaps the problem is simplifying by assuming that the potentials are evaluated at the observation point's position at time t, but with the charge's position at tr. Therefore, the potentials are expressed in terms of the distance from r to s(tr), which is d. But the options don't show this; instead, they use r. This is confusing.

Alternatively, maybe the options are not using d but |r - s(t)|, but in that case, the potentials would be evaluated at the current position of the charge, which is not correct. Therefore, perhaps the problem is using a different approach, perhaps using the Lienard-Wiechert potentials in a different form.

Alternatively, perhaps the problem is considering the charge's velocity as a constant, so the potentials are static. But the problem states that the charge is moving with trajectory s(t), so velocity is changing, implying a time-varying potential. Therefore, the potentials must involve the retarded time.

Wait, perhaps the problem is considering the potentials in a simplified form where the velocity at the retarded time is approximated as the current velocity. But I don't think that's the case here.

Alternatively, perhaps the problem is using the Coulomb potential but modified by the velocity. Wait, but for a moving charge, the potentials are more complicated. The scalar potential is still (q/(4πε₀ r')), where r' is the distance at the retarded time, but the vector potential involves the velocity at the retarded time.

Wait, but the problem mentions that the velocity is at the field generating instant tr. So the potentials at time t (t > tr) depend on the charge's position and velocity at tr. Therefore, the potentials should be expressed in terms of d and v at tr.

But the options use r and t, not d and tr. So perhaps the problem is mistyped, but according to the problem statement, the potentials are expressed in terms of r and t.

Alternatively, perhaps the problem is using the velocity at time t, but that would not be correct for a moving charge. Wait, but the problem states that the velocity is at the field generating instant tr, so the potentials should be evaluated at tr, not t. Therefore, the potentials are functions of r and tr, but the options use t instead. This is conflicting.

Given this confusion, perhaps the problem is intended to be simplified, and the potentials are expressed as if the charge were static, but with a velocity term. Alternatively, perhaps the problem is using the Coulomb potential but with an added term due to the velocity. Let's think again.

Wait, another approach: The scalar potential for a moving charge can be expressed as the Coulomb potential plus a term involving the velocity. But that's not accurate. The correct expressions involve the retarded time.

Alternatively, let's consider the Liénard-Wiechert potentials. The scalar potential is:

φ = (q / (4πε₀)) * (1 / |r - s(tr)|) * (1 + (v²/c²))

But I'm not sure. Alternatively, the scalar potential is:

φ = (q / (4πε₀)) * (1 / |r - s(tr)|) * (1 - (v·n)/c²)

where n is the unit vector in the direction of r - s(tr). But again, this might not be accurate.

Alternatively, the vector potential can be written as:

A = (μ₀ q / 4π) * (v / |r - s(tr)|) * (1 - (v·n)/c²)

But I need to check the exact forms.

Wait, according to the standard Lienard-Wiechert potentials, the scalar potential is:

φ = (q / (4πε₀)) * [1 / (|r - s(tr)| (1 - (v·n)/c²)) ]

where n is the unit vector pointing from the charge's position at tr to the observation point r. This can be rewritten as:

φ = (q / (4πε₀)) * [1 / (|d| - (v·d)/c²) ]

since n = d / |d|. Therefore, the denominator becomes |d| - (v·d)/c².

But in the options, none of the scalar potentials match this. Option D's V is qc/(4πε₀ (d c + d·v)), which is different. Option C's V is similar. So perhaps the problem is using a different sign convention or approximation.

Alternatively, perhaps the problem is using the approximation that (v·d)/c² is negligible, so the denominator becomes |d|, leading to the Coulomb potential. But that can't be, since the velocity would affect the potential.

Wait, but the problem states that the velocity is at the field generating instant tr. So maybe the velocity is considered to be constant, but that's not stated. Alternatively, perhaps the problem is using a different gauge or approximation.

Alternatively, perhaps the problem is using the Coulomb potential multiplied by a factor involving the velocity. Let's look at the options again.

Option B: V is the Coulomb potential, and A is (v/c²) V.

Wait, if the scalar potential is the Coulomb potential, then the vector potential would have a term proportional to velocity. But how?

Wait, in the case of a static charge, A is zero. When the charge is moving, the vector potential becomes non-zero. The exact expression for the vector potential in terms of the velocity involves the magnetic vector potential, which for a moving charge is (μ₀ q / 4π) * (v × r̂) / r², but this is for a static charge. For a moving charge, it's different.

Alternatively, perhaps the problem is using the approximation that the potentials are the Coulomb potential and the magnetic vector potential due to a current. But that's not the case here.

Alternatively, maybe the problem is using the potentials in a different gauge, such as the Coulomb gauge. But even then, the potentials for a moving charge would not just be the Coulomb potential multiplied by some factor.

Given the confusion, perhaps the correct answer is option B, where V is the Coulomb potential and A is (v/c²) V. But why would A be proportional to V?

Wait, let's think in terms of the relation between scalar and vector potentials for a moving charge. In the case of a static charge, A is zero. For a moving charge, the vector potential is given by:

A = (μ₀ q / 4π) * (v × r̂) / r²

But this is different from what's in the options. Alternatively, perhaps the problem is using a different expression, such as the vector potential being proportional to the current, which is charge density times velocity.

Alternatively, perhaps the problem is considering the scalar potential as the Coulomb potential and the vector potential as a correction term due to the velocity. But I'm not sure.

Alternatively, perhaps the problem is using the potentials in a different form, such as in the rest frame of the charge, but that complicates things.

Alternatively, maybe the problem is using the multipole expansion, considering that the charge is moving, so the monopole term (scalar potential) is present, and the dipole term (vector potential) is present. But for a charge moving with constant velocity, the monopole term is the Coulomb potential, and the dipole term is proportional to the velocity.

Wait, the dipole approximation for the vector potential is:

A ≈ (μ₀ q / 4π) * (v × r̂) / r²

But this is for a charge moving with velocity v at the origin. In our case, the charge is at position s(tr) at time tr, so the vector potential would involve the position s(tr). Therefore, the vector potential would be:

A ≈ (μ₀ q / 4π) * (v(tr) × (r - s(tr))̂) / |r - s(tr)|²

But this would be the dipole term. However, the problem's options don't involve a cross product of velocity and position, so perhaps this is not the case.

Alternatively, maybe the problem is using a different approach, such as the scalar potential being the Coulomb potential and the vector potential being proportional to the velocity. But I don't recall such a formula.

Alternatively, perhaps the problem is using the potentials in a specific gauge where the vector potential is related to the scalar potential via the velocity. For example, in the Lorenz gauge, the potentials satisfy certain conditions. But even then, I don't recall the exact expression.

Alternatively, perhaps the problem is using a simplified form where the potentials are expressed as:

V = q/(4πε₀ r)

A = (μ₀ q / 4π) * (v / c²) * (1/r)

But this seems arbitrary. Alternatively, perhaps the problem is using the fact that the scalar potential is the Coulomb potential, and the vector potential is the magnetic potential due to a moving charge, which is proportional to the velocity.

Wait, but the vector potential for a moving charge is not simply proportional to the velocity. It's more complicated. However, if we consider the vector potential in the static case, it's zero. When the charge moves, the vector potential can be expressed as:

A = (μ₀ q v × r̂)/(4π r²)

But this is the dipole term. However, in the options, the vector potential is expressed as a scalar multiplied by v. For example, option B has A = (v/c²) V, which would be a vector proportional to v. But in reality, the vector potential is a vector perpendicular to v and r. So unless the problem is using a specific scenario where the velocity is along a particular axis, this might not hold.

Alternatively, perhaps the problem is using a non-rigorous approach, assuming that the vector potential is proportional to the velocity. But I don't think that's correct.

Given all this confusion, perhaps the correct answer is option B. Because if the scalar potential is the Coulomb potential, then the vector potential would be proportional to the velocity. But I'm not entirely sure. Alternatively, perhaps the correct answer is option D, but I'm not sure.

Wait, another approach: Let's consider the units again. The correct expression for the scalar potential must be V = q/(4πε₀ r), which is option A and B. The vector potential in option B is (v/c²) V, which would have units (m/s)/(m²/s²) * V = (s/m) * V, which is correct for a vector potential.

But according to the Lienard-Wiechert formula, the vector potential is:

A = (μ₀ q / 4π) * (v / (c² |r - s(tr)| (1 - (v·n)/c²)) )

Which would have units of (H/m * C * m/s) / (m²/s² * m) )= (H·C·m/s) / (m²/s²·m) )= (H·C·s) / (m²) ). But H has units Wb/A·m = (V·s)/A·m. So substituting, we get (V·s/A·m * C * s) / m² )= (V·s·C·s)/(A·m³) = (V·C·s²)/(A·m³). But V = J/C = (N·m)/C. So substituting, (N·m/C * C * s²)/(A·m³) )= (N·m·s²)/(A·m³) )= (N·s²)/(A·m²) ). But N = kg·m/s², so substituting, (kg·m/s²·s²)/(A·m²) )= kg/(A·m). But A is ampere, so kg/(A·m) is not a standard unit. Therefore, perhaps the units don't match, indicating that the expression for A in the Lienard-Wiechert formula is incorrect, which is not possible. Therefore, there must be a mistake in the approach.

Given all this, perhaps the correct answer is option B, where the scalar potential is the Coulomb potential and the vector potential is proportional to (v/c²) times the scalar potential. However, I'm not entirely confident. Alternatively, maybe the correct answer is option D, but the units don't match. Wait, in option D, the scalar potential has units V/m, which is incorrect. Therefore, the only possible correct answer with correct units is option B.

But I'm not entirely sure. Alternatively, perhaps the problem is using a different gauge where the potentials are expressed differently. For example, in the Coulomb gauge, the scalar potential is the Coulomb potential, and the vector potential is zero. But when the charge is moving, the vector potential is non-zero. However, the problem states that the potentials are evaluated at time t, with the field generated at tr, so the vector potential should not be zero. Therefore, the problem must be considering the vector potential to be non-zero, which suggests that option B's vector potential is not correct.

Alternatively, perhaps the problem is using the potentials in a different way. Let's think again.

Wait, the problem mentions that the potentials are evaluated at time t, where t > tr, and the observation point r. The field generating instant tr is the time when the charge was at position s(tr). Therefore, the potentials at time t must depend on the charge's position and velocity at tr. Therefore, the potentials should be expressed in terms of d (distance between r and s(tr)) and the velocity v at tr.

Looking at the options again:

Option D: V = qc/(4πε₀ (d c + d·v)), A = μ qc v/(4π (d c + d·v))

But as we discussed earlier, the units of V in option D are incorrect. Similarly, option C's V has incorrect units. Option A and B have correct units. Therefore, perhaps the correct answer is option B, even though the vector potential doesn't match the Lienard-Wiechert formula. Alternatively, perhaps the problem is using a simplified expression where the vector potential is proportional to the velocity, even though that's not accurate.

Alternatively, perhaps the problem is considering the potentials due to the charge's motion in a way that the scalar potential is the Coulomb potential and the vector potential is a magnetic dipole term. But in that case, the vector potential would involve the cross product of velocity and position, which isn't present in the options.

Given all this confusion, and considering the units, the only option with correct units for both potentials is option B. Therefore, despite the possible discrepancy in the vector potential's form, the correct answer is likely option B.
</think>

The problem involves determining the scalar potential \( V \) and vector potential \( \vec{A} \) at a point \( \vec{r} \) due to a moving charge \( q \), where the charge's trajectory is \( \vec{s}(t) \), and the observation time \( t > t_r \) (with \( t_r \) being the retarded time). The potentials are expressed in terms of the distance \( d \) between the observation point \( \vec{r} \) and the charge's position at the retarded time \( \vec{s}(t_r) \), and the velocity \( \vec{v} \) of the charge at \( t_r \). The goal is to identify the correct expressions for \( V \) and \( \vec{A} \) from the given options.

### Key Analysis:

1. **Scalar Potential (\( V \)):**
   - The scalar potential for a moving charge is typically derived from the Liénard-Wiechert potential, which accounts for the retarded time and the velocity of the charge. The general form involves the distance \( d \) and the velocity \( \vec{v} \) at the retarded time. However, the problem simplifies the potentials to be expressed in terms of \( r \) and \( t \), not \( d \) and \( t_r \). This discrepancy complicates the interpretation.
   - The Coulomb potential \( V = \frac{q}{4\pi\epsilon_0 r} \) is a valid expression for the scalar potential when the charge is static. Since the problem does not explicitly state that the charge is static, this form might not fully account for the motion of the charge. However, none of the options explicitly include the Liénard-Wiechert-like correction terms.

2. **Vector Potential (\( \vec{A} \)):**
   - The vector potential for a moving charge also involves terms dependent on the velocity \( \vec{v} \) and the distance \( d \). The problem's options suggest simplified forms, such as \( \vec{A} = \frac{\mu_0 q}{4\pi} \frac{\vec{v}}{c^2} V \), where \( V \) is the scalar potential. This form assumes a direct proportionality between \( \vec{A} \) and \( \vec{v} \), which may not align with the full Liénard-Wiechert expressions but could represent an approximation.

3. **Option Analysis:**
   - **Option A:** \( V = \frac{q}{4\pi\epsilon_0 r} \), \( \vec{A} = \frac{\vec{v}^2}{c^2} V \).  
     - The scalar potential matches the Coulomb potential, which is correct for a static charge. However, the vector potential \( \vec{A} = \frac{\vec{v}^2}{c^2} V \) does not align with the standard Liénard-Wiechert form, as \( \vec{A} \) should depend on \( \vec{v} \) and \( d \), not \( \vec{v}^2 \).
   - **Option B:** \( V = \frac{q}{4\pi\epsilon_0 r} \), \( \vec{A} = \frac{\vec{v}}{c^2} V \).  
     - The scalar potential matches the Coulomb potential, which is correct for a static charge. The vector potential \( \vec{A} = \frac{\vec{v}}{c^2} V \) introduces a direct proportionality between \( \vec{A} \) and \( \vec{v} \), which could represent an approximation. This option has correct units for both \( V \) and \( \vec{A} \).
   - **Option C:** \( V = \frac{q c}{4\pi\epsilon_0 (d c - \vec{d} \cdot \vec{v})} \), \( \vec{A} = \frac{\mu_0 q \vec{v}}{4\pi (d c - \vec{d} \cdot \vec{v})} \).  
     - The units of \( V \) and \( \vec{A} \) are incorrect, as they involve terms like \( d c \), which mix distance and velocity dimensions.
   - **Option D:** \( V = \frac{q c}{4\pi\epsilon_0 (d c + \vec{d} \cdot \vec{v})} \), \( \vec{A} = \frac{\mu q \vec{v}}{4\pi (d c + \vec{d} \cdot \vec{v})} \).  
     - Similar to Option C, the units of \( V \) and \( \vec{A} \) are incorrect, as the denominator involves terms like \( d c \), which mix distance and velocity dimensions.

4. **Conclusion:**
   - The problem's options introduce inconsistencies in units for Options C and D. Options A and B have correct units for both \( V \) and \( \vec{A} \). However, the vector potential in Option B, \( \vec{A} = \frac{\vec{v}}{c^2} V \), does not align with the full Liénard-Wiechert expressions but could represent a simplified approximation. Given the focus on units and the lack of a clear justification for the simplified vector potential in Option B, it is the most plausible choice.

ANSWER: B