In fair weather, over flat ground, there is a downward...

In fair weather, over flat ground, there is a downward electric field of about $150 \mathrm{N} / \mathrm{C} .$ (a) Assume that the Earth is a conducting sphere with charge on its surface. If the electric field just outside is $150 \mathrm{N} / \mathrm{C}$ pointing radially inward, calculate the total charge on the Earth and the charge per unit area. (b) At an altitude of $250 \mathrm{m}$ above Earth's surface, the field is only $120 \mathrm{N} / \mathrm{C}$ Calculate the charge density (charge per unit volume) of the air (assumed constant). [Hint: See Conceptual Example $16.8 .]$

Key Concepts

Electric Field Variation with Altitude

The variation of the electric field with altitude demonstrates how the field strength changes with distance from the charged surface. The decrease in field magnitude with altitude can provide insight into the distribution of charges in the medium above the surface, serving as a basis for further calculations involving charge density in that region.

Surface Charge Density

Surface charge density is defined as the amount of charge per unit area on the surface of a conductor. In the context of a conducting sphere, after determining the total charge from the external electric field, one can calculate the charge per unit area by dividing the total charge by the surface area of the sphere.

Volume Charge Density

Volume charge density represents the amount of charge per unit volume within a material or region of space. When the electric field is known at two different altitudes, the difference between these fields can be used, via Gauss’s Law, to infer the amount of net charge present in the intervening volume assuming the charge is distributed uniformly.

Gauss's Law

Gauss's Law states that the electric flux through a closed surface is equal to the net charge enclosed divided by the permittivity of free space. This fundamental concept allows one to relate the measurable electric field to the distribution of charge, and is particularly useful when dealing with high symmetry situations such as spherical charge distributions.

Conducting Sphere Model

The conducting sphere model assumes that all the excess charge resides on the surface of the sphere and that the electric potential is constant throughout the conductor. For a spherical conductor, this implies a symmetric radial electric field outside the sphere, whose magnitude can be directly related to the total charge on the sphere using Gauss's Law.

Transcript

00:01 Well, the electric field, due to a solid sphere of radius a, date carries a charge q in the region.

00:07 External to the sphere is equivalent to date of a point charge located at the center of the sphere.

00:15 And therefore, electric field e is equal to k times q divided by r square.

00:25 Here r is greater than a r is greater than a now we are given the magnitude of electric field so magnitude of electric field is equal to 150 newton per cologne and the earth is modeled is a conducting sphere with a charge q on its surface and we know that the radius of the earth radius of earth is equal to 6 .37 multiplied by 10 to the power 6 meter 10 to the power 6 meter right this is radius of the earth okay and further now in part a we need to determine the total charge of the sphere we need to find out the total charge of the sphere so let's do let's find out total charge of the sphere so the magnitude of the electric field just outside the earth's spherical surface is found by using e is equal to k times q right divided by r e square r is the radius of the earth and from here q is equal to e times r e square e times r e square right divided by e times r e squared divided by k this is r e radius of the earth and now by substituting values we get q equals well e is 150 right so 150 into multiply by radius of earth which is 6 .37 multiplied by 10 to the power 6 whole square divided by coulamps constant which is 8 .99 multiply by 8 .99 multiply by 10 to the power 9 and therefore charge k equals 6 .77 multiplied by 10 to the power 5 colon.

03:05 Right and further now so since the electric field points towards the earth, the total charge of the earth is negative, therefore the total charge of the earth is q is equal to minus 6 .77 multiplied by 10 to the power 5 cologne...

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