Cylindrical Coordinate Systems

Questions and Answers

What does the variable $R$ represent in the context of the electric field equation?

The total electric field at point P

The position vector of the charge Q

The distance vector from charge Q to the field point (correct)

The unit vector in the direction from Q to the field point

In the equation for the electric field $E = \frac{Q}{4\pi \varepsilon |r|^3} \mathbf{a}_R$, what does the '4π' in the denominator signify?

It represents the total charge in the system.

It is a factor arising from the spherical symmetry of the electric field. (correct)

It is a scaling factor related to the geometry of the field.

It indicates the number of charges present.

When multiple point charges are present, how is the net electric field calculated?

By taking the vector sum of electric fields from each individual charge. (correct)

By taking the maximum electric field among all charges.

By averaging the individual electric fields of each charge.

By calculating the scalar sum of individual electric fields.

What is the effect of the medium's permittivity ($, \varepsilon$) on the electric field strength $E$?

It is inversely proportional to the electric field strength. Signup and view all the answers

If charge Q is positioned at the origin, what expression represents the electric field at a point r?

$E = \frac{Q}{4\pi \varepsilon |r|^3} \mathbf{a}_R$ Signup and view all the answers

What does the direction of the unit vector $a_r$ represent in spherical coordinates?

The direction of the position vector r Signup and view all the answers

Which unit vector is perpendicular to the plane formed by the z-axis and the line OP?

$a_ heta$ Signup and view all the answers

What is the mutual relationship between the unit vectors $a_r$, $a_ heta$, and $a_ ho$?

They are mutually perpendicular Signup and view all the answers

How would you describe the unit vector $a_ ho$ in relation to the other unit vectors?

It is normal to the plane formed by $a_r$ and $a_ heta$ Signup and view all the answers

What mathematical operation is critical for transforming vectors from spherical to Cartesian coordinates?

Dot product Signup and view all the answers

The angle $ heta$ describes which aspect in the spherical coordinate system?

The angle made by the position vector with the z-axis Signup and view all the answers

What describes the direction of the unit vector $a_ heta$ in the spherical coordinate system?

Perpendicular to $a_r$ in the plane of OP Signup and view all the answers

Which unit vector is in the plane formed by the z-axis and line OP?

$a_r$ and $a_ heta$ Signup and view all the answers

What is the expression for $B_\rho$ in cylindrical coordinates, given the vector $B = y a_x - x a_y + z a_z$?

0 Signup and view all the answers

In the context of the dot product, what is the result of $a_\phi \cdot a_x$?

$-\sin(\phi)$ Signup and view all the answers

What is the significance of the angle $\theta$ in spherical coordinates?

It is the angle made by the line OP with the z-axis. Signup and view all the answers

How is the vector $B$ transformed into cylindrical coordinates?

By using the relationships between Cartesian and cylindrical unit vectors. Signup and view all the answers

What is the expression for $B_\phi$ in cylindrical coordinates when transforming the vector $B = y a_x - x a_y + z a_z$?

$-\rho$ Signup and view all the answers

What does the expression $a_\rho \cdot a_y$ yield?

$\cos(\phi)$ Signup and view all the answers

What are the coordinates of point P in spherical coordinates?

r, θ, φ Signup and view all the answers

Which component of the vector $B$ is missing in the cylindrical representation?

$B_θ$ Signup and view all the answers

What is the formula for the electric field due to a line charge?

$E = \frac{1}{4\pi\epsilon} \int \frac{\rho_L , dL'}{R^3}$ Signup and view all the answers

In the context of the electric field due to surface charge, what does the symbol $ ho_s$ represent?

Charge density per unit area Signup and view all the answers

When calculating the electric field due to a volume charge, which formula is used?

$E = \frac{1}{4\pi\epsilon} \int \frac{\rho_v , dv'}{R^3}$ Signup and view all the answers

To solve for total charge in a cylinder with given charge density, which coordinate system is recommended?

Cylindrical coordinates Signup and view all the answers

The distance vector from the elemental charge to the field point is denoted as which symbol?

R Signup and view all the answers

Which of the following components is not typically considered for calculating the electric field?

Mass density Signup and view all the answers

How does the electric field due to a surface charge differ from that of a volume charge?

Surface charge incorporates area density Signup and view all the answers

In calculating the total charge in the given example, what constant value does $ ho_v$ contain?

$100 e^{-z}$ Signup and view all the answers

What is the expression for the potential due to a continuous volume charge distribution?

$V = \frac{1}{4\pi\epsilon} \int \frac{\rho_v , dv'}{|\mathbf{r} - \mathbf{r}'|}$ Signup and view all the answers

What does the equation $dV = -E , dl$ represent?

The incremental work in moving a unit charge over an incremental distance Signup and view all the answers

According to the equations, what is the relationship between the electric field $E$ and the potential $V$?

$E = - abla V$ Signup and view all the answers

What conclusion can be drawn from the equation $\nabla \times E = -\nabla \times \nabla V = 0$?

Electric field lines do not form closed loops Signup and view all the answers

Which parameter is involved in the computation for the potential due to a continuous surface charge distribution?

$\rho_s$ Signup and view all the answers

What does the breakdown of $dV$ into partial derivatives with respect to $x$, $y$, and $z$ signify?

The potential changes independently in all three dimensions Signup and view all the answers

What does $\rho_L$ represent in the context of potential due to continuous line charge distribution?

Line charge density Signup and view all the answers

In the expression $dV = \nabla V \cdot dl$, what does the term $dl$ represent?

An infinitesimal displacement in the electric field Signup and view all the answers

Study Notes

Circular Cylindrical Coordinate System

Unit vectors in cylindrical coordinates include aρ, aφ, and az.
Dot products involving these unit vectors with Cartesian coordinates:
- aρ · ax = cos(φ)
- aρ · ay = sin(φ)
- aρ · az = 0
- aφ · ax = -sin(φ)
- aφ · ay = cos(φ)
- aφ · az = 0

Vector Transformation Example

Transform vector ( B = y a_x - x a_y + z a_z ) into cylindrical coordinates:
- Components in cylindrical coordinates: ( B = Bρ aρ + Bφ aφ + Bz az )
- Calculation of components:
  - ( Bρ = 0 )
  - ( Bφ = -ρ )
Resultant vector in cylindrical coordinates: ( B = -ρ aφ + z az )

Spherical Coordinate System

Spherical coordinates defined by: ( r ) (distance from origin), ( θ ) (angle with z-axis), ( φ ) (angle in xy-plane).
Unit vectors: ar, aθ, aφ are mutually perpendicular.
ar points in the direction of the position vector, aθ is perpendicular to ar, and aφ lies in the plane formed with the z-axis.

Dot Products in Spherical Coordinates

To transform vectors between spherical and Cartesian coordinates, know dot products between unit vectors:
- Example for the plane formed by the z-axis and OP includes calculations for dot products involving ar, aθ, aφ.

Electrostatics Fundamentals

Electric field ( E ) due to a point charge at origin: ( E = \frac{Q}{4\pi ε |r|^3} a_R )
For multiple point charges, the net electric field is the vector sum of individual electric fields.

Electric Field from Continuous Charge Distributions

For line charge density: ( E = \frac{1}{4\pi ε} \int \frac{ρ_L dL'}{R^3} )
For surface charge density: ( E = \frac{1}{4\pi ε} \int \frac{ρ_S dS'}{R^3} )
For volume charge density: ( E = \frac{1}{4\pi ε} \int \frac{ρ_V dV'}{R^3} )

Example: Charge in a Cylinder

Total charge in a cylinder of height 30 cm and radius 10 cm with charge density: ( ρ_V = 100e^{-z} (x^2 + y^2) - 0.25 )
Suitable to calculate using cylindrical coordinates for volume charge distribution.

Electric Potential Due to Charge Distributions

Potential from continuous volume charge: ( V = \frac{1}{4\pi ε} \int \frac{ρ_V dV'}{|r - r'|} )
Similar formulas apply for surface and line charge distributions.

Relationship Between Electric Field and Potential

Incremental work relates electric field and potential: ( dV = -E · dl )
Expressing potential in terms of its derivatives: ( dV = \nabla V · dl ) leads to ( E = -\nabla V )
Curl of a gradient is zero, indicating that vector fields derived from potentials are conservative.

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Description

This quiz covers the essentials of the circular cylindrical coordinate system, focusing on the relationships and dot products between unit vectors in both cylindrical and Cartesian systems. Enhance your understanding of how these coordinate systems interplay and their applications in physics and engineering.