Minimum Surface Charge Density for Spark Creation

In summary, the minimum surface charge density needed for a spark to occur at the tip of a metal needle is directly related to the electric field strength, with the formula Q/ε0 = E. This means that the electric field strength must exceed 3.0x10^6 N/C for a spark to occur. The surface charge density is not uniform on the needle's surface and the tip is not flat, and the electric field lines originate from a unit positive charge.
  • #1
m84uily
33
0

Homework Statement


A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0x10^6 N/C.
What is the minimum surface charge density for producing a spark?

Homework Equations



int(EdA) = (Qin) / e0

The Attempt at a Solution



I imagine the surface as a cylinder with the electric field going parallel to it.

int(EA) = backface + side + frontface = EA + EA = 2EA
E = 3.0x10^6 N/C
A = (pi)r^2

(2)(3.0x10^6 N/C)(pi)r^2 = (Qin) / e0

(e0)(2)(3.0x10^6 N/C)(pi)r^2 = Qin

Then, I don't want r is! I've probably gone about solving this all wrong, thanks in advance for your help.
 
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  • #2
You know that free charges can not exist inside a metal so what do you mean on Qin?
The surface charge density is not uniform on the surface of a needle, and the tip of a needle is not flat.
You also might remember that Q/ε0 electric field lines originate from a unit positive charge, and the electric field strength is the number of field lines traversing a normal unit surface.

So the electric field strength given is related directly to the surface charge density on the metal.

ehild
 
  • #3
You definitely cleared up a lot for me! Thanks!

So:

Q/ε0 = E

Q = E(ε0)
 
  • #4
It is not charge (Q) but surface charge density (σ). But the result is OK.

ehild
 
  • #5


Your approach is correct. To find the minimum surface charge density, we need to find the minimum charge that will produce an electric field strength of 3.0x10^6 N/C at the tip of the needle. This can be done by rearranging the equation you have derived:

Qin = (e0)(2)(3.0x10^6 N/C)(pi)r^2

Now, we know that the charge density is defined as the charge per unit area, so we can divide both sides by the surface area of the needle (pi*r^2) to get the minimum surface charge density:

σmin = (Qin) / (pi*r^2) = (e0)(2)(3.0x10^6 N/C)

Therefore, the minimum surface charge density for producing a spark is (e0)(2)(3.0x10^6 N/C).
 

1. What is minimum surface charge density for spark creation?

The minimum surface charge density for spark creation is the minimum amount of charge required on the surface of an object in order to create a spark. This is typically measured in coulombs per square meter.

2. How is minimum surface charge density related to spark creation?

The minimum surface charge density is directly related to spark creation. As the charge density on the surface of an object increases, the likelihood of a spark being created also increases.

3. What factors affect the minimum surface charge density for spark creation?

There are several factors that can affect the minimum surface charge density for spark creation, including the material of the object, the humidity in the air, and the distance between the object and the source of the spark.

4. Is there a minimum surface charge density threshold for spark creation?

Yes, there is a minimum surface charge density threshold for spark creation. This threshold varies depending on the factors mentioned above, but generally it is around 100 coulombs per square meter.

5. How can the minimum surface charge density for spark creation be calculated?

The minimum surface charge density for spark creation can be calculated using mathematical equations that take into account the factors mentioned above. It can also be measured experimentally using specialized equipment.

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