Maxwell's Equations: How E and B Fields Interact

[MatinSAR]

Homework Statement

In the context of Maxwell's equations, are the electric field (𝐸) and magnetic field (𝐵) the same in each equation, or do they represent different fields?

Relevant Equations

Maxwell's Equations.

In Maxwell's equations, the electric field appears in both Gauss's Law and Ampère's Law. Do these refer to the same electric field, or are they different components of the overall electric field? I think Gauss's Law gives electric field 1 and Ampère's Law gives electric field 2. So the total electric field is sum of these two. And there are two different sources that create electric field, time varying magnetic field and static charges. Am I wrong?

Edit 1 :
The problem is getting bigger. There is another E field in Faraday's Law!

Edit 2:
Upon further reflection: When dealing with static charges in space, we can use Gauss's Law to determine the electric field $\vec E$. If this electric field remains constant over time, there won't be a corresponding magnetic field, and the problem is resolved. However, if the electric field changes over time, a magnetic field $\vec B$ will be generated. Currents $\vec J$ can also influence this magnetic field. If the magnetic field is time-varying, it will induce another electric field. Hence, we need to account for this induced electric field in addition to the original one.

 

[kuruman]

They are the electric field and the magnetic field at any point in space $\mathbf r$ at any time $t$. I stress "point" in that Maxwell's equation in differential form are point functions.

For example, in Gauss's law if you have two separate volume charge distributions, you would write
$$\mathbf {\nabla}\cdot\mathbf E=\frac{\rho_1}{\epsilon_0}+\frac{\rho_2}{\epsilon_0}.$$The electric field on the left hand side is the vector sum of the contributions from the two charge distributions at any point in space.

I think Gauss's Law gives electric field 1 and Ampère's Law gives electric field 2.

You are confusing fields and their sources. In very general terms, Maxwell's equations relate the fields with their sources. In the order that you have listed the equations

1. Says that the sources of diverging electric field lines from a point are charge distributions.
2. Says that the sources of diverging magnetic field lines from a point do not exist. This is another way of saying that magnetic field lines always form closed loops.
3. Says that the sources of circulating electric field lines are time-varying magnetic fields.
4. Says that the sources of circulating magnetic field lines are moving charge carriers and/or time-varying electric fields.

Note that all fields and sources are at a point in space and a point in time. Also note that by specifying the divergence and the circulation (curl) of a field at every point in space, one has specified the field in that space. One then plays the game of figuring out the electric and magnetic field in regions of space given a specific configuration of sources. In other words, I give you $\rho$ and $\mathbf J$ in all space at all times and you give me $\mathbf E$ and $\mathbf B$ everywhere in space at all times.

 

[MatinSAR]

@kuruman Thank you for your reply. I've read your post several times, but I still don't understand my initial question. Are the electric fields in each of the four Maxwell equations the same? For example, inside a capacitor, we use Gauss's Law to find the electric field due to static charges. But what happens if there's also a time-varying magnetic field?

 

[Philip Koeck]

@kuruman Thank you for your reply. I've read your post several times, but I still don't understand my initial question. Are the electric fields in each of the four Maxwell equations the same? For example, inside a capacitor, we use Gauss's Law to find the electric field due to static charges. But what happens if there's also a time-varying magnetic field?

Within the same text one symbol always stands for exactly one physical quantity.
So, for example, every vector E stands for the same thing, the electric field.

 

[renormalize]

@kuruman Thank you for your reply. I've read your post several times, but I still don't understand my initial question. Are the electric fields in each of the four Maxwell equations the same? For example, inside a capacitor, we use Gauss's Law to find the electric field due to static charges. But what happens if there's also a time-varying magnetic field?

The same unique $E$ and $B$ fields must simultaneously satisfy all four Maxwell equations.

 

[kuruman]

@kuruman Thank you for your reply. I've read your post several times, but I still don't understand my initial question. Are the electric fields in each of the four Maxwell equations the same? For example, inside a capacitor, we use Gauss's Law to find the electric field due to static charges. But what happens if there's also a time-varying magnetic field?

Perhaps you have an incomplete picture in your mind about what a field is. A field is defined with respect to a region of three-dimensional space. At every point in that region, the field has a magnitude and, if it is a vector field, a direction.

Consider for example the region of air space above a frozen lake. The air temperature will be different at different points. All the different points in the region are gathered to form "the field". For example, the temperature at point A above the frozen lake it will be lower than at point B above a fire on the shore that a camper has lit to keep warm, $T_B>T_A.$ When the camper leaves after extinguishing the fire, the temperature at point B will decrease and eventually reach the same value as $T_A$.

Note that $T_B$ is temperature-dependent while $T_A$ is not. This does not make one temperature somehow different from the other. Temperature is temperature and measured by a thermometer. Although $T_A$ may have a different value from $T_B$, it is "the same" as $T_B$ in the sense that both temperatures are parts of the same temperature field. To describe this field, we need to find a function $T(x,y,z,t)$ that gives the value of the temperature at any point $(x,y,z)$ at any time $t.$ Thus, a scalar field is a collection of scalar quantities at different points in space.

Electric and magnetic fields follow the same idea except that they are vector fields. They need a region of space where they can be defined as a collection of magnitudes and directions. In that region, the electric field at point A, $\mathbf E_A$ is "the same" as the electric field at a different point B, $\mathbf E_B$ in the sense that both belong to the same collection of magnitudes and directions. Furthermore, if one of them is time-dependent this "sameness" is not affected.

 

[MatinSAR]

Within the same text one symbol always stands for exactly one physical quantity.
So, for example, every vector E stands for the same thing, the electric field.

Good point! Thank you.

The same unique $E$ and $B$ fields must simultaneously satisfy all four Maxwell equations.

Thank you for your help.

Perhaps you have an incomplete picture in your mind about what a field is. A field is defined with respect to a region of three-dimensional space. At every point in that region, the field has a magnitude and, if it is a vector field, a direction.

Thank you for your time.


The problem is solved. Thank you to everyone for their time and assistance.