Cylindrical Coordinate System

The cylindrical coordinate system is introduced in electromagnetics to handle problems involving cylindrical symmetry, such as coaxial cables, cylindrical capacitors, long straight wires, and circular waveguides. Using Cartesian coordinates for such geometries leads to unnecessarily complex integrals. The cylindrical system aligns the coordinate surfaces with the problem geometry, making field expressions compact and integration tractable.

For GATE and university examinations, cylindrical coordinates appear frequently in problems involving coaxial transmission lines, Gauss's law with cylindrical Gaussian surfaces, and Ampere's law with circular paths around current-carrying conductors.

Core Concept Explanation

In the cylindrical coordinate system, a point P in space is located by three coordinates: rho (the perpendicular distance from the z-axis), phi (the azimuthal angle measured from the positive x-axis in the x-y plane), and z (the vertical height along the z-axis). The ranges are rho from 0 to infinity, phi from 0 to 2*pi, and z from negative infinity to positive infinity.

The three unit vectors are arho, aphi, and az. The unit vector arho points radially outward from the z-axis. The unit vector aphi is tangential, pointing in the direction of increasing phi at the observation point. The unit vector az is identical to the Cartesian az, pointing upward along the z-axis. Importantly, arho and aphi change direction as phi changes, so they are position-dependent unit vectors, unlike Cartesian unit vectors.

This position dependence of unit vectors is the key conceptual difference from Cartesian coordinates. When differentiating a vector field in cylindrical coordinates, extra terms arise from differentiating the unit vectors themselves. This is why the divergence, gradient, and curl expressions look different in cylindrical coordinates compared to Cartesian.

Mathematical Expression

The transformation from cylindrical to Cartesian coordinates is direct. Given cylindrical coordinates (rho, phi, z), the Cartesian equivalents are: x = rho cos(phi), y = rho sin(phi), z = z. The inverse transformation is: rho = sqrt(x^2 + y^2), phi = arctan(y/x), z = z.

The unit vector transformation between the two systems is: arho = cos(phi) ax + sin(phi) ay, and aphi = -sin(phi) ax + cos(phi) ay. These show explicitly how arho and aphi depend on the azimuthal position phi.

The differential volume element in cylindrical coordinates is dV = rho drho dphi dz. The extra factor of rho arises from the geometry of the cylindrical shell. When integrating over a cylindrical volume, this factor must always be included or the result will be incorrect.

The del operator (gradient) in cylindrical coordinates is: del f = (df/drho) arho + (1/rho)(df/dphi) aphi + (df/dz) az. The factor 1/rho in the phi term accounts for the fact that equal increments in phi correspond to larger arc lengths at larger rho.

Practical Understanding

Cylindrical coordinates are the natural choice for finding the electric field of an infinitely long line charge. Using Gauss's law, a cylindrical Gaussian surface of radius rho and length L is chosen. The electric flux through this surface is simply E x 2*pi*rho*L, and the enclosed charge is rho_L x L, giving E = rho_L / (2*pi*epsilon_0*rho) in the arho direction directly.

In the case of a coaxial cable, the electric field exists only in the region between the inner and outer conductors, and it points in the arho direction. The magnetic field due to the current in the inner conductor circulates in the aphi direction. Both of these are naturally expressed in cylindrical coordinates without any trigonometric complications.

Given:
Convert point P with cylindrical coordinates (rho=5, phi=60 deg, z=3) to Cartesian.
Also find the Cartesian form of unit vector arho at this point.

Why this formula applies:
Transformation between coordinate systems is needed when combining
fields expressed in different systems.

Formula:
x = rho cos(phi), y = rho sin(phi), z = z
arho = cos(phi) ax + sin(phi) ay

Substitution:
phi = 60 deg, cos(60) = 0.5, sin(60) = 0.866
x = 5 x 0.5 = 2.5
y = 5 x 0.866 = 4.33
z = 3

Calculation:
P in Cartesian = (2.5, 4.33, 3)

arho = cos(60) ax + sin(60) ay
arho = 0.5 ax + 0.866 ay

Verification:
|arho| = sqrt(0.5^2 + 0.866^2) = sqrt(0.25 + 0.75) = sqrt(1) = 1

Final Answer:
Cartesian point: P = (2.5, 4.33, 3) m
Unit vector arho = 0.5 ax + 0.866 ay (magnitude confirmed = 1)

Exam Tip: In GATE, the most common error with cylindrical coordinates is forgetting the rho factor in the volume element dV = rho drho dphi dz. Always check whether you are integrating over a cylindrical volume and include rho before integrating. Also remember that arho and aphi depend on phi, not on rho or z.

• Cylindrical coordinates (rho, phi, z) are used for geometries with axial symmetry such as wires, cylinders, and coaxial cables.
• Unit vectors arho and aphi are position-dependent and change direction with phi. az is always constant and identical to Cartesian az.
• Transformation to Cartesian: x = rho cos(phi), y = rho sin(phi), z = z.
• Differential volume: dV = rho drho dphi dz. The rho factor must never be omitted during volume integration.
• Position vector in cylindrical: r = rho arho + z az. There is no phi term in the position vector itself.

Quick Revision

• Three coordinates: rho (radial), phi (azimuthal), z (axial). Ranges: rho>=0, 0<=phi<2pi, z any value.
• Unit vectors: arho x aphi = az, aphi x az = arho, az x arho = aphi. Right-handed system.
• Transformation: x = rho cos(phi), y = rho sin(phi). Inverse: rho = sqrt(x^2+y^2), phi = arctan(y/x).
• Unit vector arho in Cartesian: cos(phi)ax + sin(phi)ay. Unit vector aphi: -sin(phi)ax + cos(phi)ay.
• Differential volume: dV = rho drho dphi dz. Differential surface (curved): dS = rho dphi dz arho.
• Position vector: r = rho arho + z az. Note: there is no explicit phi component in the position vector.
• Key GATE trap: writing dV = drho dphi dz without the rho factor is a very common and costly error in integration problems.