Constants in scalar and vector potentials

In summary, the conversation discusses a scalar potential and a vector potential, both of which involve the constants ##a## and ##\gamma##. These constants represent the amplitude and attenuation factor, respectively, in the formulas. They have physical meaning and can be measured through the electric and magnetic fields. However, additional conditions (known as gauge conditions) must be used in order to accurately measure them.
  • #1
struggling_student
9
1
We have a scalar potential $$\Phi(\vec{r})=\frac{q}{4\pi\epsilon_0} \left( \frac{1}{r} - \frac{a^2\gamma e^{-\gamma t}\cos\theta}{r^3}\right)$$

and a vector potential $$\vec{A}(\vec{r})=\frac{a^2qe^{-\gamma t}}{4\pi\epsilon_0r^4}\left(3\cos\theta\hat{r} + \sin\theta\hat{\theta} \right) .$$

how do I interpret the constants ##a## and ##\gamma##. Do they have any physical meaning or are they arbitrary, unmeasurable values?
 
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  • #2
Focusing first on the formula for vector potential, we can say that ##a## is part of the amplitude while ##\gamma## is the attenuation factor (or the damping factor) with respect to time. In time ##t=\frac{5}{\gamma}## the amplitude loses 99.32% of its initial value at time t=0.

Similar things can be said for the ##\frac{1}{r^3}## term of the scalar potential.

As to if they are measurable things, yes they are. At least from what I know is that usually we can measure the scalar potential (not sure about the vector potential) and from that we can infer the values of a ang gamma.

P.S We can measure both scalar and vector potential but we have to use additional conditions (known as gauge conditions, e.g. Coulomb gauge, Lorentz gauge. In any case what we actually can measure is electric and magnetic field , ##\vec{E},\vec{B}##.
 
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Related to Constants in scalar and vector potentials

1. What are scalar and vector potentials?

Scalar and vector potentials are mathematical functions used to describe the properties of a physical field. Scalar potentials are used to describe the magnitude of a field, while vector potentials describe the direction and strength of the field.

2. What is a constant in scalar and vector potentials?

A constant in scalar and vector potentials is a value that remains unchanged throughout the entire field. It is used to simplify the mathematical equations and make them more manageable.

3. How are constants determined in scalar and vector potentials?

Constants in scalar and vector potentials are determined through experimental data and mathematical calculations. They are chosen to fit the observed behavior of the physical field and to make the equations more accurate.

4. What is the significance of constants in scalar and vector potentials?

Constants in scalar and vector potentials play a crucial role in understanding and predicting the behavior of physical fields. They help us simplify complex equations and make accurate predictions about the behavior of the field.

5. Can constants in scalar and vector potentials change?

Yes, constants in scalar and vector potentials can change depending on the physical conditions of the field. For example, the constants may change if the temperature or pressure of the field changes. However, they are typically chosen to remain constant for a specific set of conditions.

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