A uniformly charged ring of radius 10.0 cm has a total charge of 75.0μC . Find the electric fiel

A uniformly charged ring of radius 10.0 cm has a total...

A uniformly charged ring of radius 10.0 cm has a total charge of 75.0$\mu \mathrm{C}$ . Find the electric field on the axis of the ring at (a) $1.00 \mathrm{cm},$ (b) $5.00 \mathrm{cm},(\mathrm{c}) 30.0 \mathrm{cm},$ and $(\mathrm{d}) 100 \mathrm{cm}$ from the center of the ring.

Key Concepts

Uniform Charge Distribution

This concept refers to the distribution of charge such that it is evenly spread over a geometric object. In the context of a ring, every element of the ring carries the same amount of charge per unit length, which simplifies calculations by allowing the use of symmetry and integration to determine the electric field at any point in space.

Electric Field on the Axis of a Charged Ring

For a uniformly charged ring, the electric field at points along the central axis can be derived by integrating the contributions from each infinitesimal charge element of the ring. The resulting expression typically involves the ring's total charge, the distance from the center along the axis, and the radius of the ring, and it encapsulates how the field strength diminishes with distance.

Symmetry in Electric Field Calculations

Symmetry is a key concept in electromagnetism that is used to simplify the computation of electric fields. In a charged ring, symmetry ensures that the lateral components of the electric field from opposite charge elements cancel each other, leaving only the axial component. This greatly simplifies the integration process and forms the basis of deriving the standard formula for the field on the axis of the ring.

Limiting Cases in Electric Field Distributions

This concept involves analyzing the behavior of the electric field derived from a charge distribution in different regions, such as very close to the ring or far away from it. In the case of a charged ring, far from the ring the field approximates that of a point charge, whereas near the ring the detailed geometry plays a significant role. Evaluating these limits provides insight into the validity of approximations and the physical behavior of the system across different scales.

Transcript

00:01 Okay, so this is the equation for the electric view of the uniform ring with the charge, which is equal to k -e -x over a square plus x -square to the power of three over two times charge cube.

00:12 A is a radius, which is given as 10 .0 centimeter, which is 0 .1 meter.

00:18 And x is the distance from the center of ring.

00:21 And in question a, it's given as 1 .0 -0 centimeter, which is 0 .01 meter.

00:26 And k .e is a coolant constant, which is 8 .99 times 10 .2 .9, newtons.

00:31 Square per kolon square and chart q is given at 75 .0 microcourt which is 75 times 10 to 96cccccccccc so let's plug in all these values back into the equation and we have electric field is equal to oh sorry 8 .99 times 10 to power 9 newton times meter per coolon square and then times 0 .01 meter over a square which is uh 0 .1 meter square plus 0 .01 meter and square, then to the power of 2, 3 over 2.

01:25 Okay? and then has charged q, which is 75 times 10 to the power of negative 6 cologne.

01:36 And this will give us open.

01:40 The electric view for question a is about 6 .64 times 10 to the the power of 6 newton per cologne.

01:54 And for question b, we use the same equation.

01:58 By this time, the x, which is the distance from the center of ring, is given as 5 .0 centimeter.

02:04 If we convert meter, it's 0 .05 meter.

02:06 So let's try to determine the electric view for this case, which is e is equal to 8 .99 times 10 to the power of 9 newton per times meter square per colon square...

Please give Ace some feedback