A solid metal sphere with radius 0.450 m carries a net...

Name: Problem 14, Chapter 22: University Physics with Modern Physics
Uploaded: 2019-03-06T22:36:45-08:00
Duration: 3 min 41 s
Channel: Cody Johnston
Description: A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) at a point 0.100 m outside the surface of the sphere and (b) at a point inside the sphere, 0.100 m below the surface.

A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) at a point 0.100 m outside the surface of the sphere and (b) at a point inside the sphere, 0.100 m below the surface.

Key Concepts

Spherical Symmetry

Spherical symmetry implies that the electric field at any point outside a uniformly charged sphere depends only on the distance from the center, allowing the charged sphere to be treated as if all the charge were concentrated at its center. This simplification is crucial when applying Gauss's Law to calculate the electric field in both the regions outside and inside the sphere.

Electrostatics of Conductors

In electrostatics, conductors have the property that their excess charge resides entirely on the surface. This means that the electric field inside a conductor in equilibrium is zero. For a metal sphere, even if it is charged, the field inside the material (and within any cavity inside the conductor) cancels out due to the redistribution of charges, which is essential in explaining why the electric field inside is zero.

Gauss's Law

Gauss's Law is a fundamental principle that relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful for calculating electric fields when the charge distribution has symmetry, such as spherical symmetry, by allowing the evaluation of the electric field based on the enclosed charge divided by the permittivity of free space.

Transcript

00:01 This is chapter 22 problem 14 from sears and zamansky's university physics.

00:06 A solid metal sphere with radius 0 .450 meters carries a net charge of 0 .250 nanoprude.

00:14 Find the magnitude of electric field, a, at a point, 0 .10 ,000 meters outside the surface of the sphere, and b, at a point inside the sphere, 0 .100 meters below the surface.

00:25 All right, the first thing i've done is i have drawn a model of the sphere.

00:32 So we're given that the radius of the sphere is 0 .450 meters, which i've denoted with capital r for the radius.

00:39 And then i'm looking at some distance 0 .10 meters outside of the sphere, represented by d for that distance.

00:47 And we're given that the total charge is 0 .250 nanoculums, but that's the same thing as 0 .250 times 10 to the negative.

00:55 Nine coolum, so i just substituted in for nano, so that way i'm in standard si units.

01:02 And the basic way of solving this expression is to use the equation that's the definition for the electric field due to a point particle, kq over r squared.

01:14 K is our electrostatic constant, 8 .99 times 10 to the 9 newt meter squared per coolum from coolum's law.

01:21 Q is going to be the total charge, that total charge of the sphere.

01:26 Well, as i just said, we were given that.

01:30 And lowercase r is the distance from the center.

01:34 So we can assume since we're outside of the sphere, we can model this as if the entire charge of the sphere was concentrated at the center.

01:41 So that means the total distance from the center of the sphere is simply the radius of the sphere, capital r, plus the distance away from the sphere, d.

01:50 So our distance away little r is the same thing as big r plus d.

01:55 So i made this substitution into the expression.

01:58 And then i've just plugged in the numbers...

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