Predict/Calculate The Electric Field of the Earth The Earth...

Predict/Calculate The Electric Field of the Earth The Earth produces an approximately uniform electric field at ground level. This electric field has a magnitude of 110 $\mathrm{N} / \mathrm{C}$ and points radially inward, toward the center of the Earth.(a) Find the surface charge density (sign and magnitude) on the surface of the Earth. (b) Given that the radius of the Earth is $6.38 \times 10^{6} \mathrm{m},$ find the total electric charge on the Earth. (c) If the Moon had the same amount of electric charge distributed uniformly over its surface, would its electric field at the surface be greater than, less than, or equal to 110 $\mathrm{N} / \mathrm{C}$ ? Explain.

Predict/Calculate The Electric Field of the Earth The Earth produces
an approximately uniform electric field at ground level. This electric
field has a magnitude of 110 $\mathrm{N} / \mathrm{C}$ and points radially inward, toward
the center of the Earth.(a) Find the surface charge density (sign and
magnitude) on the surface of the Earth. (b) Given that the radius of
the Earth is $6.38 \times 10^{6} \mathrm{m},$ find the total electric charge on the Earth.
(c) If the Moon had the same amount of electric charge distributed
uniformly over its surface, would its electric field at the surface be
greater than, less than, or equal to 110 $\mathrm{N} / \mathrm{C}$ ? Explain.

Key Concepts

Scaling of Electric Field with Radius

When a total electric charge is distributed uniformly over the surface of a sphere, the electric field at the surface depends inversely on the square of the radius of the sphere. This concept demonstrates how changes in size (or scale) of an object affect the intensity of the electric field generated by a given amount of charge.

Electric Field

The electric field is a vector field that represents the force per unit charge experienced by a positive test charge placed in a region. It is fundamental to understanding how charges interact with their surroundings and is central to the study of electromagnetism.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the total charge enclosed by that surface. It is particularly useful for calculating electric fields when there is a high degree of symmetry in the charge distribution, as it simplifies the integration process.

Surface Charge Density

Surface charge density is defined as the amount of electric charge per unit area on a surface. It is a key parameter in determining the electric field near the surface of a conductor or any charged object, especially when applying Gauss's Law to relate the charge distribution to the resulting field.

Spherical Symmetry in Electrostatics

Spherical symmetry implies that the charge distribution is uniform over a sphere, which allows the electric field to be calculated easily due to the uniformity in all radial directions. This symmetry simplifies the use of Gauss's Law by reducing complex integrals into straightforward algebraic expressions.

Transcript

00:01 Question 90 states that the earth produces an approximately uniform electric field at ground level.

00:07 This electric field has a magnitude of 110 nons per coulum and points radially inward towards the center of the earth.

00:15 Part a of the question states that we want to find the surface charge density, sign, and magnitude on the surface of the earth.

00:22 So we'll just do part a first.

00:25 So based on what's given to us, i establish that the electric field is negative.

00:29 That's simply because remember if we're saying we're on top of the earth, the electric field pointing downward towards the center of the earth, then whatever charge that we're dealing with has to be negative.

00:39 So, of course, the electric field in this scenario would therefore be negative.

00:44 So if you want to find the electric field for our object, well, we know the equation for that.

00:47 It would be k, we call the charge of the earth capital q, over the radius of our earth squared.

00:53 This is, of course, the equation.

00:55 However, we don't know what our charge is.

00:59 So we can represent that our charge if we're trying to solve for surface area.

01:04 Well, the charge is just the surface area, surface density, sorry, sigma multiplied by the surface area of our object, again, over our distance term squared.

01:21 Our area, our surface area of a sphere, of course, our area is 4 pi r squared over r squared, which would mean our radius terms cancel out.

01:40 And if i expand what i have for my k value, so i have sigma 4 pi on our 9 .5.

01:46 Numerator of course on our denominator the value 4k is 1 over 4 pi epsilon not therefore our 4 pi terms also cancel out and we're left from an expression for electric field to simply be surface charge density over epsilon not so if we want to solve for surface charge density we can simply take our electric field and multiply by epsilon not plugging in our value we would find that the surface charge density of the earth here again has to be negative because of the negative electric field of the earth is negative 9 .7 times 10 to negative 10 coulums per meter squared.

02:32 Great, let me just section this off so don't blur into other sections.

02:36 Part b states that given the radius of the earth is 6 .38 times 10 to the 6 meters, find the total electric charge of earth.

02:46 So as we know, what we kind of use in our first part of our question, we can represent the charge as the surface charge density, multiplied by the surface air...

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