Some planetary scientists have suggested that the planet...

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $3.63 \times 10^{16} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{C}$ at the planet's surface. Calculate: (a) the total electric charge on the planet; (b) the electric field at the planet's surface (refer to the astronomical data inside the back cover); (c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.

Key Concepts

Surface Charge Density

Surface charge density is the measure of electric charge per unit area on a surface, providing insight into how charge is distributed over an object. It is particularly useful when the charge is assumed to be uniformly spread over a surface, allowing for calculations that link total charge and electric field characteristics.

Electric Field

The electric field represents the force per unit charge experienced by a small test charge placed in the vicinity of other charges. It describes how charges influence each other in space and is calculated by the charge and its distribution, especially in symmetric cases where it can be derived using Gauss’s Law.

Electric Flux

Electric flux measures the number of electric field lines passing through a given surface and is directly linked to the charge distribution that produces the field. It is a key quantity in applying Gauss’s Law, as it simplifies complex electric field configurations when symmetry is present.

Gauss's Law

Gauss's Law is a fundamental principle in electrostatics that relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the net flux is equal to the enclosed charge divided by the permittivity of free space, providing a powerful method to calculate charges when symmetric fields, like those of spherical objects, are involved.

Transcript

00:01 So we're at mars and we have a net electric flux and it's pointing directly in.

00:04 The first question wants to know, well, okay, given that we have a net electric flux pointing inwards, what is the total electric charge on the planet? so here we go.

00:17 Remember that the total flux, 5 equals total charge inside of a region divided by some constant epsilon not.

00:25 So simply multiply epsilon not to the other side to get that q.

00:30 Which is the charge, equals 5 times epsilon not.

00:37 If i was given in the problem, so we just plug it in.

00:40 Here we go.

00:43 3 .63 times 10 to the 16 times epsilon, which is a constant.

00:56 You just look it up.

00:57 8 .85 times 10 to minus 12 to get a glorious answer of about 3 .2 times 10 to the 5th qom.

01:10 Okay, now what about the sign? well, remember that our net electric flux was pointing directly in.

01:17 Remember that negative charges have an electric field that goes towards them.

01:22 So that means that this is actually a negative answer.

01:25 Negative charge.

01:28 Okay, part b.

01:30 Given that we have this net electric flux and we have a certain charge on the planet, let's go ahead and find the electric field at the planet's surface.

01:41 Okay? so, you can either use goust's law, which says that you have the integral of the electric field times small area element equals total charge divided by epsilon 9.

01:54 This is actually phi right here.

01:56 You can use goust's law, or you can remember that whenever you have a point charge, it's electric field e equals k, that's just constant, q, that's the total charge inside, divided by r squared.

02:14 I'm going to go ahead and use this because it's easier...

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