What Happens to a Point Charge Inside a Uniformly Charged Sphere?

[danilorj]

Homework Statement

Suppose a strength that acts between two charges depends on the distance, 1/r^a, where
1) a>2
2)a<2
What will happen with a punctual charge, if it is put within a uniformly charged sphere. At the initial moment the punctual charge is in rest.

Homework Equations

F is proportional to 1/r^a.

The Attempt at a Solution

The solution says that for a> 2 the punctual charge q goes toward point O, that it is the center of the sphere, in case of same signals.
and for a<2 the punctual charge goes toward point B that is in the surface of the sphere.
And if the signals are opposite it happens the opposite too.
I don't understand this. Why of going toward the center and the surface of the sphere?

 

[tiny-tim]

hi danilorj!

(btw, we don't say a "punctual charge", we say a "point charge" …

"punctual" means not early and not late, for an appointment )

do an integration, of the force from a small area dθdφ, over the whole sphere …

what do you get? ​

 

[danilorj]

oh man thanks for the warning.. it is point charge.

But still don't understand what to do with this force. Is it necessary to calculate? The problem states that is proportional to 1/r^a, I don't know about this 'a' whether it can be less than one or even negative.

 

[tiny-tim]

hi danilorj!

(just got up :zzz:)

The problem states that is proportional to 1/r^a, I don't know about this 'a' whether it can be less than one or even negative.

(try using the X² button just above the Reply box )

just leave it as "a", and do the integration …

the result will be a formula using "a" ​

 

[Nickie]

I would first clarify the terms being used in this question. A punctual charge refers to a point charge, which is a theoretical concept in which a charge is assumed to be concentrated at a single point in space. A uniformly charged sphere means that the charge is evenly distributed throughout the entire sphere, rather than being concentrated at a single point.

Based on this information, I would interpret the question as asking what would happen to a point charge if it is placed inside a uniformly charged sphere.

For the first case where a>2, the force between the two charges would decrease as the distance between them increases. This means that the force between the point charge and the charges in the sphere would be strongest at the center of the sphere, where the distance between them is the shortest. Therefore, the point charge would be pulled towards the center of the sphere.

For the second case where a<2, the force between the two charges would increase as the distance between them decreases. This means that the force between the point charge and the charges in the sphere would be strongest at the surface of the sphere, where the distance between them is the shortest. Therefore, the point charge would be pulled towards the surface of the sphere.

The direction in which the point charge would be pulled would also depend on the charges being attracted or repelled. If the charges have the same sign, they would be attracted towards each other and the point charge would move towards the center or surface accordingly. If the charges have opposite signs, they would be repelled from each other and the point charge would move away from the center or surface accordingly.

In summary, the behavior of the point charge within a uniformly charged sphere would depend on the strength of the force between them, which is determined by the distance between the charges and the value of the exponent a.