# Magnetic field from displacement currents in a capacitor, and an applied exterior magnetic field

If an exterior magnetic field ($B$) is applied to oppose or support the magnetic field produced by the displacement currents ($B_D$), what would happen to the electric field within the gap? And the capacitor in general?

Diagram:

The exterior magnetic field($B$)is not uniform, as indicated, also varying with time.

• How would that external magnetic field be applied? Commented Aug 5, 2018 at 12:10
• Added a diagram to the question, hope that clarifies. If you meant by what's causing the magnetic field? Could be anything(From magnet, solenoid, single wire). Commented Aug 5, 2018 at 15:01

In general, the only practical way to produce a magnetic field supporting or opposing the magnetic field produced by the displacement current (presumably, in all points in space), is to generate another current spatially aligned with the original current and flowing in the same or the opposite direction.

The spatial alignment means that this current will have to flow in the leads of the capacitor and, therefore, either increase or decrease the rate of its charging and discharging and, with it, its electric field.

• What if the field was covering "encompassing" the whole capacitor's interior? And it's also not uniform(added a diagram in the question) wouldn't it cause and effect? Also, I'm very curious as to why the spatial alignment has to be with the leads/plate to plate (i.e within the capacitor). Commented Aug 5, 2018 at 15:03
• @e.d.m The magnetic field you drew has completely different shape/direction than the field produced by displacement current, so it won't specifically support or oppose it, therefore, we can consider its effect on the electric field independently of the effect of the magnetic field due to displacement current. If this new field is changing, it will produce circular electric field in the plane of the page, which will just superimpose on the electric field due to the displacement current, i.e., the electric field at any given point will be a sum of the two fields.
– V.F.
Commented Aug 5, 2018 at 15:45
• @e.d.m The requirement for spatial alignment would be necessary if you wanted to scale the existing field up or down (to the extent that it could be completely canceled) without changing its shape.
– V.F.
Commented Aug 5, 2018 at 15:47
• I appreciate the feedback. What's mostly confusing is the consequences; firstly, if the exterior magnetic field was aligned and opposing the displacement current's magnetic field, or if the exterior magnetic field was varying and producing an electric field that of which opposes the electric field within the capacitor, how would that effect the capacitance or charging/discharge of a capacitor. Besides spatially aligning. Commented Aug 5, 2018 at 17:41
• @e.d.m I think so. Of course, the external magnetic field would have to be in sync with the normal charge/discharge current - otherwise the effect of the external field would change with time, i.e., sometimes it would support the normal current and sometimes it would oppose it.
– V.F.
Commented Aug 6, 2018 at 22:27