FInd a simple expression for the vector potential which will yield this field

[Nusc]

em

Homework Statement

In a certain region, the magnetic field as a linear function of time is given by

$$B = Bo \frac{t}{/tau} z hat$$
Bo and tau constants.

A)FInd a simple expression for the vector potential which will yield this field.
B)Assuming the scalar potential is a constant, find E from the above result for the vector potential. Check that your answer is consistent with the diff form of faradays law

Homework Equations

$$\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$

$$\mathbf{A} = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}(\mathbf{rsingle-quote},t_r)}{\cal{R}} d\tausingle-quote$$

$$t_r \equiv t - \frac{\cal{R}}{c}$$

The Attempt at a Solution

$$B= \nabla \times A$$

We take the curl of B on the LHS and on the RHS we have
$$\nabla \times (\nabla \times A )$$

Eventually we will get to this $$\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$This is assuming were in the Lorentz Gauge, we don't know if we are. Or do we assume we are? We have to satisfy E = -delV -dA/dt but we don't know either of those quantities.

 

[Nusc]

say if

kt y-hat = K x xhat

What mathematical operation would I apply to show that
K = kt z-hat

I don't thin kthat is even relevent

 

[pam]

A) Take curl(BXr)/2 and see what you get.

 

[nrqed]

Homework Statement

In a certain region, the magnetic field as a linear function of time is given by

$$B = Bo \frac{t}{/tau} z hat$$
Bo and tau constants.

A)FInd a simple expression for the vector potential which will yield this field.
B)Assuming the scalar potential is a constant, find E from the above result for the vector potential. Check that your answer is consistent with the diff form of faradays law

Homework Equations

$$\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$

$$\mathbf{A} = \frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}(\mathbf{rsingle-quote},t_r)}{\cal{R}} d\tausingle-quote$$

$$t_r \equiv t - \frac{\cal{R}}{c}$$

The Attempt at a Solution

$$B= \nabla \times A$$

We take the curl of B on the LHS and on the RHS we have
$$\nabla \times (\nabla \times A )$$

Eventually we will get to this $$\nabla^2 \mathbf{A} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$


This is assuming were in the Lorentz Gauge, we don't know if we are. Or do we assume we are? We have to satisfy E = -delV -dA/dt but we don't know either of those quantities.

The simples way to go about this is to simply write explicitly $$B= \nabla \times A$$
in cartesian coordinates.

You will get simple equations for A_x, A_y and A_z. It should be easy to pick functions A-x, A_y, A_z that satisfy those equations. You don't need anything else.

 

[Nusc]

The simples way to go about this is to simply write explicitly $$B= \nabla \times A$$
in cartesian coordinates.

You will get simple equations for A_x, A_y and A_z. It should be easy to pick functions A-x, A_y, A_z that satisfy those equations. You don't need anything else.

Doing this would only yield

dA_y/dx - dA_x/dy = Bo(t/tau) since B has only z-components.

Solving this differential equation would still give unknown A_y A_x.

There should be an explicit way of doing this, no?

 

[genneth]

Remember that the vector potential is not physically different even if you add a gradient of a function. So you have a certain freedom available to pick the solution.

 

[nrqed]

Doing this would only yield

dA_y/dx - dA_x/dy = Bo(t/tau) since B has only z-components.

Solving this differential equation would still give unknown A_y A_x.

There should be an explicit way of doing this, no?

Of course A is defined up to gauge transformations only so many different A's give the same physics. This is why the question asks you to find a simple expression for A. Just pick the simplest A that yields the correct B

(don't forget to make sure that the other components of the equation also work)

 

[Nusc]

OKay problem solved. You can close this thread.