At what point on the x-axis is the field greatest?

In summary, a circular ring with a radius r and a positive uniformly spread electric charge is centered at the origin and lies in a plane perpendicular to the x-axis. The electric field in the x-direction, E, at any point x on the x-axis is given by E= kx/(x^2 + r^2)^(3/2) for k>0. The critical points are found to be zero, indicating that the electric field inside the ring is zero. The charge on a conductor resides on its outer surface, so the greatest electric field is found outside the ring and the least is inside the ring. It is not necessary to take the derivative for this conceptual problem.
  • #1
mugzieee
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a circular ring of wire radius r lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-direction, E, at the point x on the x-axis is given by E= kx/(x^2 + r^2)^(3/2) for k>0. At what point on the x-axis is the field greatest? Least?

After taking the first derivative i ended up with:
E' = k(-2x^2 + r^2 - 3xr)/(x^2 + r^2)^(5/2)
i know after the derivative I am supposed to find the critical points then classify them and find the global min and global max, but for critical points i end up with only a zero, waht does it look like is wrong here?
 
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  • #2
The Important thing to know is that , The charge on a conductor ( in this case the circular ring ) resides on its outer surface. Therefore, electric field E inside the ring is zero.

2nd, For a point on the charged spherical conductor or outside it, the charge may be assumed to be concentrated at its centre.

Based on this least E which is zero, inside the ring.

The greatest is outside, the ring.

I hope this helps.

I don't know if you really need to take the derivative in this question. This seems more like a conceptual problem.
 
  • #3


The critical point found from taking the derivative is correct, as it is the only point where the derivative is equal to zero. However, this does not necessarily mean it is the only critical point. To determine if it is a maximum or minimum, you need to take the second derivative and evaluate it at the critical point. If the second derivative is positive, then the critical point is a minimum, and if it is negative, then it is a maximum. In this case, taking the second derivative would result in a positive value, indicating that the critical point is a minimum. Therefore, the point on the x-axis where the electric field is greatest is at x = 0, and the point where it is least is at the point where the electric field is equal to zero, which can be found by setting the original equation equal to zero and solving for x.
 

1. What is the meaning of the x-axis in relation to the field?

The x-axis represents the horizontal distance in a given coordinate system. In the context of a field, it could represent the distance from a source of the field, such as a charged particle or a magnet.

2. How is the field measured on the x-axis?

The field is typically measured in units of force per unit charge, such as newtons per coulomb for electric fields or teslas for magnetic fields. The field at a specific point on the x-axis can be measured using a field strength meter or by calculating the field using mathematical equations.

3. Can the field on the x-axis be negative?

Yes, the field on the x-axis can be negative. This indicates that the field is pointing in the opposite direction of the positive x-axis. For example, if the field is negative at a certain point on the x-axis, it means that the force exerted on a positive charge would be in the negative x-direction.

4. How does the distance on the x-axis affect the strength of the field?

The strength of the field on the x-axis is inversely proportional to the square of the distance from the source of the field. This means that as the distance on the x-axis increases, the field strength decreases. This relationship is described by the inverse square law.

5. How can I find the point on the x-axis where the field is greatest?

The point on the x-axis where the field is greatest can be found by calculating the field at different points along the x-axis and comparing the values. Alternatively, if the field is created by a known source, such as a point charge, the maximum field can be found using mathematical equations and principles of symmetry.

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