- #1
Sturk200
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We are given a uniformly charged (non-conductor) cube. It is required to understand how the field strength along the edges relates to the field strength over the center of a face.
The correct answer is apparently that the field will be weaker along the edges than over the center of a face, but I am having trouble seeing why.
I convinced myself that the field will be stronger along the edges by the following argument.
First simplify the picture and talk about a square sheet of charge of of length 2a. I want to place a point a distance .5a directly above the center of the sheet and look at the field contribution at that point from an infinitesimal piece of the sheet which is located a distance a from the center (i.e. in the middle of one of the edges). The answer depends on two things: the distance of the charge from the point of interest, which in this case is Sqrt(1.25a^2), and the angle made between the vertical direction and the field line, which here is about 63.4 degrees.
Note that due to symmetry we are interested only in the vertical component, so as the angle gets larger and as the distance gets larger, the field gets smaller.
Now consider the alternative case. We bend the sheet down the center at a right angle and form two sides of a box. Now (like before) we explore a point located .5a directly above the edge of the box, at a 45 degree angle from the extension of either of the sides (i.e. measured radially out from the edge). And we want to find the field contribution at this point due to the same infinitesimal element from before; i.e. due to a point a distance a from the right angle bend, in the middle of one of the edges. Now since the sheet is bent we are interested not in the vertical component of field, but rather in the component in the direction parallel to a line drawn radially out from the corner of the bend, all other components being canceled by symmetry. So how does this radial component of field compare to the vertical component from the previous scenario?
It seems like bending the sheet has two effects: it increases the distance from the point of interest to the element of charge whose influence we are considering (which weakens the field); and it decreases the angle between the field line and the expressed component (which strengthens it). The question is, which of these effects is stronger? On doing some trig, I found that the new distance is about 1.4a and the new angle is about 30.4 degrees, which gives the result that the field due to this element in the bent case is about a tenth stronger than the field due to the analogous element in the flat case. If this is true for one element, I reason that it ought to be true for every element on the sheet.
In sum, bending the sheet increases the distance between charge elements and a point of interest some fixed distance away from the sheet, but it also decreases the proportion of the field component that is canceled by symmetry. It seems that when the sheet is bent by 90 degrees, the second effect outweighs the former by a small amount, the conclusion being that the field at the edge of the cube should be stronger than at the center of a face.
Now numerous people have told me that this result is wrong. I want to believe them. But I also want to know where my reasoning is wrong. Any help is very much appreciated.
Thanks.
The correct answer is apparently that the field will be weaker along the edges than over the center of a face, but I am having trouble seeing why.
I convinced myself that the field will be stronger along the edges by the following argument.
First simplify the picture and talk about a square sheet of charge of of length 2a. I want to place a point a distance .5a directly above the center of the sheet and look at the field contribution at that point from an infinitesimal piece of the sheet which is located a distance a from the center (i.e. in the middle of one of the edges). The answer depends on two things: the distance of the charge from the point of interest, which in this case is Sqrt(1.25a^2), and the angle made between the vertical direction and the field line, which here is about 63.4 degrees.
Note that due to symmetry we are interested only in the vertical component, so as the angle gets larger and as the distance gets larger, the field gets smaller.
Now consider the alternative case. We bend the sheet down the center at a right angle and form two sides of a box. Now (like before) we explore a point located .5a directly above the edge of the box, at a 45 degree angle from the extension of either of the sides (i.e. measured radially out from the edge). And we want to find the field contribution at this point due to the same infinitesimal element from before; i.e. due to a point a distance a from the right angle bend, in the middle of one of the edges. Now since the sheet is bent we are interested not in the vertical component of field, but rather in the component in the direction parallel to a line drawn radially out from the corner of the bend, all other components being canceled by symmetry. So how does this radial component of field compare to the vertical component from the previous scenario?
It seems like bending the sheet has two effects: it increases the distance from the point of interest to the element of charge whose influence we are considering (which weakens the field); and it decreases the angle between the field line and the expressed component (which strengthens it). The question is, which of these effects is stronger? On doing some trig, I found that the new distance is about 1.4a and the new angle is about 30.4 degrees, which gives the result that the field due to this element in the bent case is about a tenth stronger than the field due to the analogous element in the flat case. If this is true for one element, I reason that it ought to be true for every element on the sheet.
In sum, bending the sheet increases the distance between charge elements and a point of interest some fixed distance away from the sheet, but it also decreases the proportion of the field component that is canceled by symmetry. It seems that when the sheet is bent by 90 degrees, the second effect outweighs the former by a small amount, the conclusion being that the field at the edge of the cube should be stronger than at the center of a face.
Now numerous people have told me that this result is wrong. I want to believe them. But I also want to know where my reasoning is wrong. Any help is very much appreciated.
Thanks.