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This is a question from my high school physics class years ago. I'm embarrassed to say I still don't understand the answer.

Say we have a non-conducting (like garnet) spherical shell, with -1 C charge spread evenly over the outer surface. At the center of the shell is a +2 C charge. What is the direction of the electric field just outside the outer surface?

Picture:

Diagram showing nonconducting sphere (gray), negative charge on surface, (blue), positive charge at center (red), and location where we are interested in the field (green)

Now, I know how to answer this in terms of Gauss's law: net positive charge inside the sphere, the field points outwards. I also know how to answer it in terms of integrating Coulumb's law: same answer. Or, you could use the fact that electric charge is the divergence of the electric field; the field should reduce in strength by half as you pass from inside the sphere to outside the sphere, so, again, same answer.

But I also know that "field lines flow out of positive charges and into negative charges". So what's going on? Are there really supposed to be field lines flowing out of the negatively charged surface?

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When you draw field lines, each one is supposed to represent a certain amount of flux. So each magnetic field like is a certain number of Webers and each electric field line is a certain number of Volt-meters.

So the number of field lines coming out of a region is proportional to the charge. So if you draw four field lines coming out of the center, then two need to stop on the shell. If you draw two million field lines coming out of the center then one million need to stop on the shell.

Basically, each unit of charge has a certain number of field lines coming out or going in. Terminating or originating.

So have half go through and half terminate. Then the correct number terminate and originate at each place.

Are there really supposed to be field lines flowing out of the negatively charged surface?

Yes, there really is a flux of field lines going through the surface

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The moment you talk about field lines, it has become a question based on field theory. The idea that field lines begins at a positive charge and terminate at a negative charge is valid only for idealized point charges; at a point with a finite charge, the charge density would be infinite; it becomes a singularity when the field concept is no more applicable. The method of Gaussian surface and Gauss's law is correct. The Gaussian surface is also an idealized concept - it does not "contact" any real physical charge.

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Field lines are a visual aid used to give some information about a field.
Faraday assigned properties to field lines and actually "counted" them to give an indication of the strength of a field.
The terminology which is still used eg flux is an indication that thinking of fields in terms of field lines produced a reasonable theory of electromagnetism and the "counting" was done by many university professors well into the twentieth century.

Field lines are a figment of the imagination and there are times when the limitations of their use are exposed but that does not mean that you should not draw pictures with field limes on them to help visualise a situation provided you understand the limitations.

For your diagram above you can draw a few radial lines from the $+2$ charge assuming that the charge is uniformly distributed within the sphere as the electric field at the centre would then be zero.
Electric field lines flowing out of positive changes and into negative charges is just a help for you to visualise or draw the field lines.
So if you were really fussy those radial lines emanating from the $+2$ charge would not be starting at the centre of the sphere but at a positive charge within the sphere. Thus the flux of field lines as you moved closer to the centre would decrease - the field was becoming weaker.

Gauss tell you that the field lines are pointing radially outwards.
In terms of charge distribution.
On the inside of the $-1$ shell there is a charge of $-2$ and on the outside of the $-1$ shell (this is the net charge on the shell) there is a charge of $+1$.

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