Showing E.dl is 0 - Why cylindrical coordinates?

• emhelp100
In summary: Rectangular coordinates are simpler, but you can also calculate the circumference in polar coordinates.
emhelp100

Homework Statement

A point charge +Q exists at the origin. Find $\oint$ $\vec{E}$ $\cdot \vec{dl}$ around a circle of radius a centered around the origin.

The Attempt at a Solution

The solution provided is:
$\vec{E} = \hat{\rho}\frac{Q}{4\pi E_0a^2}$
$\vec{dl}=\hat{\phi}\rho d\phi$
Why are cylindrical coordinates being used here?

I think these are spherical coordinates. Why do you think they are cylindrical?

In principle, one can use any coordinate system for this problem. However, it's simpler to use spherical coordinates in this case. To see why, write ##\vec E## and ##\vec {dl}## in rectangular (Cartesian) coordinates.

jtbell said:
In principle, one can use any coordinate system for this problem. However, it's simpler to use spherical coordinates in this case. To see why, write ##\vec E## and ##\vec {dl}## in rectangular (Cartesian) coordinates.
Not really sure what it would be in rectangular coordinates...

Start by drawing a diagram that shows ##\vec E## and ##\vec {dl}## at a point a distance a from the origin and at some arbitrary angle φ with respect to the x-axis. Then resolve both of those vectors into x- and y- components.

emhelp100 said:

Homework Statement

A point charge +Q exists at the origin. Find $\oint$ $\vec{E}$ $\cdot \vec{dl}$ around a circle of radius a centered around the origin.

[snip]

Why are cylindrical coordinates being used here?

As @jtbell states, any coordinate system can be used.
You'll get the same result... but one choice of coordinates might be easier than another.
Generally speaking, if you choose a coordinate system that exploits a symmetry in the problem, then your math problem (your integration problem) will be simpler.
The point charge at the origin describes something with spherical symmetry.
The circle centered around the origin has cylindrical symmetry.

(Would you rather calculate the circumference of a circle in rectangular coordinates or in polar coordinates?)

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used to describe positions in three-dimensional space. They consist of a radius, an angle, and a height, and are often used in problems involving cylindrical or circular shapes.

2. How are cylindrical coordinates different from Cartesian coordinates?

Cylindrical coordinates use different variables to describe position compared to Cartesian coordinates. In cylindrical coordinates, the variables are radius, angle, and height, while in Cartesian coordinates, the variables are x, y, and z. Additionally, cylindrical coordinates are more suited for describing cylindrical or circular shapes, while Cartesian coordinates are better for rectangular shapes.

3. Why is it important to show that E.dl is 0 in cylindrical coordinates?

In electromagnetism, E.dl (electric field dotted with a differential length) is used to calculate the work done by an electric field on a charged particle. In cylindrical coordinates, the electric field is often not constant, so it is important to show that E.dl is 0 in order to accurately calculate the work done and understand the behavior of the electric field.

4. How is E.dl calculated in cylindrical coordinates?

E.dl is calculated by taking the dot product of the electric field vector and the differential length vector. In cylindrical coordinates, the electric field vector is expressed as (Er, Eθ, Ez), and the differential length vector is expressed as (dr, rdθ, dz). The dot product of these two vectors is then integrated over the desired path to find the total work done by the electric field.

5. Can other coordinate systems be used to show E.dl is 0?

Yes, other coordinate systems such as spherical coordinates or even Cartesian coordinates can be used to show that E.dl is 0. The specific choice of coordinate system depends on the problem at hand and which system will make the calculations and analysis easier.

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