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# About surface charge density on a plate coonected to a terminal of a battery

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#### belax

##### Member level 2
Imagine we have two items.

1)A "U" volts battery with two terminals "+" and "-" (e.g. 12V car battery).
2)Thin copper plate of size AxB. It is neutrally charged - number of electrons equal to protons.

I touch a "minus" terminal of a battery with this plate (plus terminal left floating).
The main question of this thread is: What is a surface charge density of plate? (after some instant - when an equilibrium is reached)?

I need this to calculate the static electric field at some distance from the plate
I don't understand how and where to start from.

Thank you.

#### KlausST

##### Super Moderator
Staff member
Hi,

I don´t think it makes sense to talk about everything floating.

I recommend to connect one of the battery terminals to earth GND as 0V reference.

Klaus

#### FvM

##### Super Moderator
Staff member
Two possible scenarios.

1. The battery is shielded by a conductive case connected to the minus terminal (e.g. a car chassis). There will be zero charge transferred to the plate and zero field.

2. The battery is not shielded and placed in the vicinity of the plate. You need to analyze the battery shape and surface voltage distribution exactly. Needs 3D field solver, beyond the scope of a simple exercise problem.

#### belax

##### Member level 2
Thanks for beginning.
Ok. The battery is not shielded and placed out of the vicinity of the plate.
Let's assume that there is a wire (thin) long enough (e.g >10 meters) that connects battery and a plate. Plate is small compared to the wire length (e.g 100x100 mm).

I have 3D solver and I know how to use it, but there must be something simple involving math and physics laws: Poisson equations, Gauss law and other laws from Maxwell set of equations. I just don't know how to start.

You also can substitute battery to a charged (somehow sometime somewhere) capacitor. I don't think it is important here.

To KlausST: in a space there is no ground.

#### FvM

##### Super Moderator
Staff member
Ok. The battery is not shielded and placed out of the vicinity of the plate.
Then there's no field.

#### belax

##### Member level 2
To FvM: Why? Do you mean no field from plate? That means if we have a test charge close to the plate this charge feels no force. right?

Then, I think it contradicts this thought experiment (for simplicity in vacuum):
Imagine you have two plates connected by wire with plus and minus terminal. Plates are initially far from each other (as in our case above). According to what you say there is no field around the plates.
Now we start move plates close to each other till they are parallel with small distance "d" between each other.

As we all know this a parallel plate capacitor and the field between plates equal to E=U/d and charge Q=(U*eps0*A*B)/d, where A=B=100mm in our case, U=voltage (potential difference) of a battery

So how could charge and field appear if there were no charge and field before (according to your claimn)?

I honestly want to understand the physics behind my first post, so I ask to support your answers by theory.
Of course I might be wrong.

#### FvM

##### Super Moderator
Staff member
You say the battery + terminal is floating and the battery is far from the plate. So why should there be charge transfer to the plate or any field generated?

#### belax

##### Member level 2
Does it (what you say) mean that the distribution of charges in the plate is the same before and after you touch the battery minus terminal with wire (which is connected to the plate)?

Endolith

### Endolith

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#### FvM

##### Super Moderator
Staff member
Yes, under the prerequisites you have stated.

#### belax

##### Member level 2
My answer is in the picture below.

Please give me a theory support of your statements.

Endolith

### Endolith

Points: 2
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#### FvM

##### Super Moderator
Staff member
You have sketched a completely different setup. Now the positive battery terminal is exposed to the outside.

The "floating terminal" point in your previous setup unfortunately involves an ambiguity that the battery and terminals must either assumed with zero size or you have to consider the exact battery shape and surface charge distribution.

As previously mentioned, shielding the battery would be a non-ambiguous way to avoid calculating the battery surface charge.

#### belax

##### Member level 2
Excuse me, but i don't get it.

The length of the wire in my first setup (and in the last setup) is such that the battery looks like a point from the view of the plate.

#### FvM

##### Super Moderator
Staff member
Relevant parameter is the capacitance between positive battery terminal and plate. Either it's zero, than it can be ignored and the plate isn't charged. Or it's nonzero, than the plate gets charged when connecting to the negative battery terminal. You need to know the capacitance to calculate the surface charge.

You define the conditions.

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