# Electrostatics problem involving a Cone

• Tanya Sharma
In summary, to take a test charge q from infinity to the apex of a cone made of an insulating material with a slant length L, the potential at the tip is calculated by adding potentials due to rings of charges with charges distributed along the surface. The surface area of the ring is given by 2π√(L'2-x2)dx.
Tanya Sharma

## Homework Statement

A cone made of insulating material has a total charge Q spread uniformly over its sloping surface. Calculate the energy required to take a test charge q from infinity to apex A of cone. The slant length is L.

## The Attempt at a Solution

Potential at the tip of the cone is calculated by adding potentials due to rings of charges .

Let us consider a ring having charge 'dq' at a distance 'x' from the tip and at slant height L'. L'=xsecθ where θ is the half angle of the cone .

The surface area dS of the ring = 2π√(L'2-x2)dx

dq=Q(dS)/(πRL)

V = ∫dV = ∫kdq/L'

This doesn't give me correct answer .I am not sure if I have approached the problem correctly.

I would be grateful if somebody could help me with the problem.

#### Attachments

• cone.JPG
4 KB · Views: 586
Tanya Sharma said:
The surface area dS of the ring = 2π√(L'2-x2)dx

Hello, Tanya. How "wide" is the ring on the surface of the cone? Is it dx?

ehild

TSny said:
How "wide" is the ring on the surface of the cone? Is it dx?

Hi TSny

The width is dxsecθ .That gives correct answer .

But I am having difficulty in comprehending (rather visualizing ) the difference between using dx and dxsecθ :shy:

My tiny brain can't differentiate between the two .I think if I use 'dx' then I am essentially covering the surface area of a cylinder ,not a cone . Am I right ?

But how is surface area of ring given by (Perimeter)(slant height) ? What is the shape of the differential element ? Is the ring in the form of a cylinder or is it more like a frustum of a cone?

ehild said:

ehild

'x' is distance of the center of the ring from the tip and L' is distance between the tip and a point on the circumference of the ring.

Last edited:
You can help your brain by imagining a much bigger θ. Clearly no cylinder, hence the secθ.

Tanya Sharma said:
'x' is distance of the center of the ring from the tip and L' is distance between the tip and a point on the circumference of the ring.

No need the centre of the ring mix in. The charge is distributed along the surface.
You get a circular section when you unroll the side of the cone. The centre of the circle is the tip of the cone. Can you see how to get the distance of a surface point from the tip? Do you see how to get the area of a ring?

ehild

#### Attachments

• cone.JPG
10.7 KB · Views: 990
1 person
ehild said:
No need the centre of the ring mix in. The charge is distributed along the surface.
You get a circular section when you unroll the side of the cone. The centre of the circle is the tip of the cone. Can you see how to get the distance of a surface point from the tip? Do you see how to get the area of a ring?

ehild

Thanks ehild

Please have a look at the attachment .I have edited your diagram .Have I represented the ring (in red) correctly ?

#### Attachments

• cone.JPG
19.1 KB · Views: 749
That "ring" is not a ring really but part of the cone surface. You do not need x at all. Let be l the distance of a point on the surface of the cone from the tip. The unrolled cone has the central angle Φ=2πr/L. Then the surface element is lΦ dl. The area of the side of the cone is A=L2Φ/2.

ehild

#### Attachments

• cone2.JPG
17.8 KB · Views: 733
Hello TSny...

Could you please respond to post#4 and post#7 .

I am still not very clear how dxsecθ is the width of the ring.I understand it is more of a calculus concept . But if you could give your views , that would be very helpful .

Does the attached diagram help? If you "unwrapped" the strip off of the cone, it would be approximately a rectangle of width as shown in the diagram and length equal to the circumference of the circular cross section of the cone at the location of the strip.

#### Attachments

• cone strip.png
1.4 KB · Views: 549
It goes like this.
Since u assumed an elementry part over the sloping surface , u must be inegrating over the surface .
So,
Instead of using dS= 2π√(L'2-x2) dx
U must use dS=2π√(L'2-x2) dl.
Hope u understood. This will give u correct answer.

## 1. How does the shape of a cone affect electrostatics?

The shape of a cone can affect electrostatics in several ways. First, the electric field lines near the pointed end of a cone will be more concentrated and intense compared to the flatter end. This is because the curvature of the cone causes the electric field to be more perpendicular to the surface, leading to a higher electric field strength. Additionally, the shape of a cone can also impact the distribution of charges on its surface, which can affect the overall electric field in and around the cone.

## 2. How can I calculate the electric field at a point near a charged cone?

To calculate the electric field at a point near a charged cone, you can use the formula for electric field intensity (E = kQ/r^2) and take into account the distance from the point to the cone's tip, as well as the angle between the point and the cone's axis. Additionally, you may need to use calculus to integrate the contributions of different portions of the cone's surface to the overall electric field at the point.

## 3. Is the electric field inside a charged cone zero?

No, the electric field inside a charged cone is not zero. While the electric field may be weaker at the center of the cone compared to the outer edges, it is still present. This is because the electric charges on the surface of the cone create an electric field that extends into the cone's interior.

## 4. How does the charge distribution on a cone affect its electric field?

The charge distribution on a cone can greatly impact its electric field. For example, if the cone is uniformly charged, the electric field lines will be more evenly distributed and the electric field will be more uniform. However, if there is a higher concentration of charge on one part of the cone compared to others, the electric field will be stronger in that area and weaker in others.

## 5. Can a charged cone attract or repel other charged objects?

Yes, a charged cone can attract or repel other charged objects, depending on the polarity of the charges. If the cone has a positive charge, it will repel other positively charged objects and attract negatively charged objects. Conversely, if the cone has a negative charge, it will attract positively charged objects and repel negatively charged objects.

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