# Variation of induced electric field with distance

Let's say we have a cylindrical uniform magnetic field which is varying with time at a constant rate. As the distance from axis increases, induced electric field or electromotive force (EMF) varies as

My questions:

1. Why does EMF changes with distance from central axis at all? Why does it care that how far it is from central exis since its uniform?
2. How can EMF be induced outside the cylinder since there is no magnetic field outside?
3. Why does it vary in this fashion? This graph is similar to variation of electric field with distance from centre of sphere of uniformly distributed charge or a planet's gravitation. How is this situation analogous to those?
• What do you mean by a “cylindrical uniform magnetic field”? Apr 8, 2018 at 5:31
• Infinitely long cylindrical region of space that has a non zero value of magnetic field in a uniform distribution which is varying with time at a constant rate. Apr 8, 2018 at 8:39

1. Why does EMF changes with distance from central axis at all? Why does it care that how far it is from central exis since its uniform?

Generally, given the magnetic field as function of position and time, the Maxwell equations imply magnitude and direction of curl of electric field inside. This by itself does not imply that electric field will increase as a function of distance from the center axis.

But almost universally a specific boundary condition is implicit - the field is zero at infinity and it has to copy the symmetry of the system generating the field (surface current on a cylinder). So electric field lines will be circles concentric with the cylinder and the only way such field can have constant curl (has to be constant due to uniform change of magnetic field inside) is when electric field strength is proportional to radius.

Another way to look at this result is that the field is due to charges and currents inside the wall. The electric field direction and strength are partially determined by distance from the current element, partially by intensity and direction of the change in the current (acceleration of the charged particles).

Then the electric field has to be zero on the center axis, because contributions from elements of the wall are of equal strength, but they have all possible directions equally often, so they cancel each other. If the point is off the axis, the symmetry is disturbed and total electric field is no longer zero. The farther the point is from the axis, the closer to some wall elements and farther from others, so the greater the electric field.

1. How can EMF be induced outside the cylinder since there is no magnetic field outside?

There is magnetic field outside in any realistic approximation of this ideal system, like when we have tightly wound long solenoid.

However, it is true that in the limit of infinite cylinder with uniform surface current throughout the surface, the magnetic field outside is zero, even if the current changes in time so magnetic field changes in time inside.

In that case, the magnetic field is zero but electric field does not have to be. There is no law that requires presence of magnetic field in place where induced electric field is. The Faraday law only requires that if there is a curl of electric field, magnetic field has to change in time. But in out case, induced electric field outside the cylinder has zero curl. The curl is nonzero only inside the cylinder.

1. Why does it vary in this fashion? This graph is similar to variation of electric field with distance from centre of sphere of uniformly distributed charge or a planet's gravitation. How is this situation analogous to those?

Not really much similar. In our case, we have changing magnetic field inside due to changing electric current on the cylinder surface; this is unsustainable process, sooner or later the process of change of current and magnetic field will stop or reverse. In the sphere example you mention, the situation is static, the field will remain indefinitely, since nothing is changing.