# Force by a changing magnetic field on a stationary charge

I already know that a moving charge experiences a force in a constant magnetic field governed by the Lorentz force formula.

What I want to know is that whether a changing magnetic field exhibits a force on a stationary electric charge?

The same question is answered in Does a changing magnetic field impart a force on a stationary charged particle?, but all the answers use the differential forms of the equations which I don't understand.

As an example, let's say that a uniform but increasing with time magnetic field exists into the screen and I place an electron in the region. What will be the the direction of the force and it's magnitude of the variation between magnetic field and time is given.

EDIT 2:

The magnetic field exists in a cylindrical region and the electron is located at a position which is above the center of the cross sectional circle.

• Perhaps Faraday's law? – K_inverse Nov 13 '18 at 11:25
• @K_inverse Faraday's Law tells me about emf induced in a object. How does it work on point charges? – Harshit Joshi Nov 13 '18 at 11:30
• If you have a specific scenario in mind then you should specify that. Right now your question is a duplicate of the one you reference, and the differential equations you say you don't understand aren't even the main focus of the accepted answer there. We want our questions to be applicable to others, so I'm not sure if the criteria for a question being a duplicate includes what the OP does/doesn't understand from other questions (I could be wrong though) Be more specific if you have something to be more specific about :) – BioPhysicist Nov 13 '18 at 12:20
• @AaronStevens I have made an edit however it now might be marked as a homework question and closed. – Harshit Joshi Nov 13 '18 at 12:31

Your formulation of the situation,

Let's say that a uniform but increasing with time magnetic field exists into the screen and I place an electron in the region. What will be the the direction of the force and it's magnitude of the variation between magnetic field and time is given.

is not sufficient to fully determine the configuration. The presence of an increasing uniform magnetic field going into the screen is enough to tell you that there is a non-conservative electric field in the plane of the screen, but it is not enough to fully specify what that electric field will look like.

More specifically, the Maxwell equations tell you that this electric field will have a certain amount of circulation: that is, the line integral of the electric field around a closed loop $$C$$ will be given by some constant, $$\oint_{\mathcal C} \mathbf E \cdot \mathrm d \mathbf r = \mathrm{const}(\mathcal C),$$ where the constant $$\mathrm{const}(\mathcal C) = A(\mathcal C) \times \frac{\mathrm dB}{\mathrm dt}$$ is given by the area $$A(\mathcal C)$$ enclosed by $$\mathcal C$$ times the rate of increase of the magnetic field. In more intuitive language, this means that if you draw a loop and look at the components of $$\mathbf E$$ along the loop (i.e. the work per unit charge performed by the field over that loop), they will add to a nonzero number proportional to the area of the loop. This is, of course, a direct break with the behaviour in electrostatics, where the electric field is conservative.

However, the Maxwell equations only give you the circulation, but they don't give you enough boundary conditions to know how to split that sum up into components, so you don't have enough information about the field to find $$\mathbf E(\mathbf r)$$. It is possible to supplement your specification of the configuration with additional information (say, with behaviour at a boundary, or with a symmetry center about which the electric field needs to be rotationally symmetric) which will allow you to find $$\mathbf E(\mathbf r)$$ at all points, but the need for that additional information is basically symptomatic that you can't isolate the situation you've given from the global picture (say, a pair of Helmholtz coils located at positions $$X$$ and $$Y$$ with a ramping current starting at time $$T$$) that produced the specific aspects you've given.

One such example is your added information,

The magnetic field exists in a cylindrical region and the electron is located at a position which is above the center of the cross sectional circle

which implicitly requires the electric field to be cylindrically symmetric about the symmetry axis of the region. This can then be solved to show that the electric field is given by $$\mathbf E(\mathbf r) = \frac{B_0}{2T}\mathbf r \times \hat{\mathbf z}= \frac{B_0}{2T} (y,-x,0)$$ assuming that the $$z$$ axis is the center of the cylinder and that the magnetic field $$\mathbf B(\mathbf r,t) = \frac{t}{T} B_0 \, \hat{\mathbf z}$$ increases linearly with time.

And, with that information, since your particle is on the symmetry axis, it will experience zero force from this field $$-$$ as required by symmetry, since the particle can't "know" which way it should get pushed. If the particle is displaced from the axis of symmetry, then it will experience a force that's orthogonal to its separation vector from that axis of symmetry.

• What else do I need to specify for fully determining the configuration? – Harshit Joshi Nov 13 '18 at 12:50
• I edited the question again. Does it specify everything now? – Harshit Joshi Nov 13 '18 at 13:12
• I also added an image where the electron is located at the point P. You mentioned a vector orthogonal to the separation vector. Which vector would that be as there could be many vectors orthogonal to a single vector? – Harshit Joshi Nov 13 '18 at 13:27
• Yes it is given in the answer as being a circle centered at the z axis. I was wondering whether the z unit vector will be into the plane or out of the plane. I believe it will be into the plane as the direction of the the magnetic field is into the plane implying that the electric field in my case will be anticlockwise in nature. Am I correct? – Harshit Joshi Nov 13 '18 at 13:35
• Actually sir, can I please ask how did you derive the function $\vec{E}(\vec{r})$. I used the equation $\int E \cdot dl = -\frac{d\phi_B}{dt}$ and was able to get the same magnitude for $\vec{E}(\vec{r})$. But how did you get the cross product between $\vec{r}$ and $\hat{z}$?? – Harshit Joshi Nov 13 '18 at 13:58