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This term is really bothering me a lot. While explaining the radial direction of electric field of a uniformly charged sphere, my book writes:

Notice the use of argument of symmetry. There is no unique direction other than the perpendicular line to the surface & that is readily the radial direction.

I couldn't comprehend the argument. Then I read the Berkeley Physics book by Edward M. Purcell; here also, unfortunately, he mentioned the same argument:

Because of spherical symmetry, the electric field at all points must be radially directed_ no other direction is unique.

Can anyone please tell me what actually is unique direction? How is it related to symmetry?

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  • $\begingroup$ I have also seen this question; mine isn't a duplicate of this. In many more places, apart from spherical symmetry, I got the term unique direction. I just want to know what actually it is & how it is related to symmetry. $\endgroup$
    – user36790
    Jun 19 '15 at 6:15
  • $\begingroup$ Related: physics.stackexchange.com/q/441005/2451 and links therein. $\endgroup$
    – Qmechanic
    Nov 15 '18 at 9:32
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The object for which you need to find the electric field is a uniformly charged sphere. Uniformly charged means that at every point on the sphere the charge density is same.

Suppose someone blind-folded you and then he rotated the sphere in some arbitrary fashion about the origin(assuming your sphere has origin as the center). Then he takes off the blind fold.

The question is: is there any way you can determine if he rotated the sphere or not by measuring its charge density at each point or some geometrical property of the sphere?

The answer is no.

Thus as the electric field is determined by the charge density ( this includes information about both the charge at every point and the geometrical distribution of charge) in space and this charge density has spherical symmetry, do you think it is intuitively reasonable for the electric field to have this symmetry too?

Now consider a spherical Gaussian surface around the sphere centered at the origin. Consider two different points on the Gaussian surface. If the electric field at these two points are different then this asymmetry should come from the distribution. But as the charge distribution has no such asymmetry they should be the same.

About the direction of the field:

Assume the field has some direction that is at some angle with the radial direction. Remember my example where I showed that any arbitrary rotation should be undetectable.

Draw a triangle on the sphere. Start from one of the vertices of the triangle and rotate the sphere along the triangle such that the current position of some point on the triangle should be the position of the vertex at $t=0$ at every instant of rotation. Make one complete rotation along the triangle.

The constraint that the rotation should be undetectable will give you the direction of Electric Field at every point on the triangle. When you complete one rotation and come back to the vertex you started with you will see that the direction you assigned earlier is incompatible with the direction you gave initially. This will be so unless the direction was radially outward.

Thus you cannot assign a unique direction for electric field other than the radial one which is compatible with spherical symmetry.

enter image description here

In the above picture you start from the top vertex. Observe two different directions for the field after one rotation.

I am sorry if I was a bit clumsy at explaining it or if I was really confusing.

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  • $\begingroup$ Yes, I could understand why the magnitude would be same but am not understanding your triangle-argument. Can you please clarify that? $\endgroup$
    – user36790
    Jun 19 '15 at 8:09
  • $\begingroup$ The essential idea of the triangle argument is to first assume there is a direction other than the radial one. Then you try to assign directions to other points on the sphere using the constraint of spherical symmetry. $\endgroup$ Jun 19 '15 at 9:01
  • $\begingroup$ Suppose you know the direction of electric field at one point say 'A'. Now to find the direction of electric field at any other point 'B' you rotate the sphere such that the position of point 'B' is the old position of point 'A'. Now spherical symmetry tells you that both the initial and final configurations of the system should be indistinguishable. Thus you can infer the direction of electric field at 'B'. In its new position the direction of electric field of 'B' should be same as the direction for old 'A'. $\endgroup$ Jun 19 '15 at 9:02
  • $\begingroup$ So there is no consistent unique assignment of direction of electric field obeying the constraints of spherical symmetry other than the radial direction. $\endgroup$ Jun 19 '15 at 9:04
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    $\begingroup$ Try imagining the assignment of directions as sticking arrows at that point. Finally when you finish the loop you will see that the arrow you need to stick at the initial point is in a different direction. May be this triangle thing is too confusing. Try imagining it. Better still try doing it yourself. Let me see if I can cook up a diagram or a video $\endgroup$ Jun 19 '15 at 9:04
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The "unique" here is, IMO, not a good, evocative word. A better one would be preferred direction or privileged direction.

Another way of looking at this is all directions are equivalent.

A further confusing subtlety is that there is also something else that the author is assuming without telling you.

There are no privileged directions for a problems with spherical symmetry. Give me a sphere with no marks on it and no other specifications, and I cannot uniquely define any direction.

Now when the author talks about the "unique" direction, he or she actually means implicitly "unique, given a particular point on the sphere". Now specify a point on the sphere. The surface then defines the direction normal to surface, i.e. the radial line. But there is no way I can uniquely define an orthogonal direction given only a sphere and a given point. I can rotate the sphere about that radius and the problem still looks the same: any orthogonal direction can be transformed into any other by an isometry.

A way to encode all this mathematically is that any isometric transformation that interchanges only equivalent directions does not change the problem. In your spherically symmetric problem, this means all homogeneous (leaving the origin fixed) isometries: i.e. reflexions about any plane through the origin or rotations about any axis through the origin. For a cylinder, on the other hand, such an isometry would a (1) reflexion about the plane orthogonal to the axial direction and cutting the cylinder in half, (2) any reflexion about any plane containing the cylinder's axis (assuming, to be pedantic, we have a circular cylinder) or (3) any rotation about the cylinder's axis.

So, solutions must be invariant to any such isometry. If you rotate your co-ordinates system about any axis through the origin for your spherical problem, the untransformed solutions must remain solutions because the problem "still looks the same".

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