(II) A solid metal sphere of radius 3.00 m carries a total charge of -5.50 μC . What is the magn

(II) A solid metal sphere of radius 3.00 m carries a total...

Key Concepts

Gauss's Law

This fundamental law of electromagnetism states that the net electric flux through a closed surface is proportional to the enclosed charge. It is extensively used to compute electric fields when the system exhibits high symmetry, such as spherical symmetry, by choosing an appropriate Gaussian surface, which simplifies the integration process.

Electric Field of a Conducting Sphere

For a conductor in electrostatic equilibrium, the free charges reside on the surface, resulting in no electric field inside the material. Outside the sphere, the field behaves as if all the charge were concentrated at the center, following an inverse-square law with distance.

Electric Field of a Thin Spherical Shell

A thin spherical shell exhibits similar behavior to a conducting sphere regarding the external field: at any point outside the shell, the electric field is as though the entire charge were concentrated at its center. However, unlike a conductor, the field inside the shell is exactly zero because of the symmetrical distribution of charge on its surface.

Electric Field of a Uniformly Charged Nonconducting Sphere

For a nonconducting sphere with charge spread uniformly throughout its volume, the electric field outside the sphere also behaves like that of a point charge located at the center. Yet, inside the sphere, the field increases linearly with the radial distance from the center because the amount of enclosed charge increases with the cube of the radius, resulting in a dependence on r that is directly proportional inside the sphere.

Spherical Symmetry

Spherical symmetry in electrostatics allows simplification of the problem by ensuring that the electric field at any point on a Gaussian surface centered on the charge distribution has the same magnitude. This symmetry greatly simplifies the application of Gauss's Law and the analysis of electric field behavior in both conductive and nonconductive materials.

Transcript

00:01 So we have a solid metal sphere which has a charge of minus 5 .5 microcurems.

00:14 We want to find the electric field at a different set of distances, first being 0 .250 meters.

00:24 Now let's look at this case first.

00:28 Because this is a metal sphere, all of its charge will be residing on the surface.

00:34 And because it's a metal, the field inside would be zero.

00:44 Hence, inside for any distance less than three meters from the center, the field will be zero in this whole region.

00:54 So 0 .250 meters is inside the sphere and hence electric field will be zero.

01:02 For part b, we are looking at 2 .90 meters, which is somewhere here, which is still within the spear, and hence the electric field is zero.

01:14 But for part c, the distance is 3 .1 meters, which is outside the sphere.

01:23 Now, there is another thing to note.

01:26 Spherical charge distributions act as if the whole charge was concentrated at the center for objects outside the sphere.

01:35 That is, at this point, it will be as if the whole charge of minus 5 .5 micropoleum was concentrated at this center.

01:46 Hence, the electric field would be simply q by 4 pi r square epsilon not, just like a point charge.

01:57 But notice that we are simply finding the magnitude of charge.

02:01 The direction will simply be radial, that is in this direction, towards the center of the spear.

02:09 So the magnitude will be charge will be 5 .5 into 10 power 6, columns.

02:21 I'm sorry, there is another pi here for the pi.

02:26 Now 1 by 4xel or not has the value of 8 .98 into 10 power 9, newton meter square per column square, if you remember, divided by r squared is 3 .1 meter square.

02:42 This gives us 5 ,140.

02:48 And notice the units, we lose a column here, meter square there, and the units are newton per column as you expect for electric field.

02:57 Similarly for part d we are looking at distance 8 meters.

03:03 So again it's as if we have a charge of minus 5 .5 microculems and at 8 meters distance we are finding the electric field.

03:17 So the electric field is again simply q by 4xelon 0 r square.

03:24 The charge is again 5 .5 into 10 power minus 6, 1 by 4x4 by epsilon not has the value of 8 .98 into 10 power 9 newton meter square per coolum squared divided by 8 meters squared.

03:40 This gives us 772 newton per column.

03:44 In fact, you could not, you need not have done this.

03:47 Now that you know this is 5 ,140 at 3 .1 meters, we know that here we divided by 3 .1 squared to get the field at 8 meters, we would have to divide instead by 8 squared.

04:03 So you could simply multiply this by 3 .1 by 8 whole squared and you would have the same answer.

04:11 Now for part e, we want to consider what happens if it was a thin metal shell instead.

04:18 The answer is nothing else, nothing different.

04:20 Even if it was a thin metal spherical shell, the charge would still be on the outside and the charge the field inside would still be 0.

04:29 And hence the all the values would be the same.

04:33 But the next part is a little interesting...

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