Spherical Coordinate System

The spherical coordinate system is used in electromagnetics when the problem has spherical symmetry, such as the field of a point charge, a conducting sphere, or a dipole antenna. In these situations, expressing fields in spherical coordinates leads to the simplest possible form of the solution. The entire Coulomb's law result, for example, expresses E as a function of r alone in the radial direction, which is impossible to write compactly in Cartesian coordinates.

GATE electromagnetics problems frequently involve spherical Gaussian surfaces, spherical charge distributions, and far-field antenna patterns, all of which are handled most efficiently in the spherical coordinate system.

Core Concept Explanation

In the spherical coordinate system, a point P is described by r (the radial distance from the origin), theta (the polar angle measured from the positive z-axis, ranging from 0 to pi), and phi (the azimuthal angle measured from the positive x-axis in the x-y plane, ranging from 0 to 2*pi). The combination of these three coordinates uniquely locates any point in three-dimensional space.

The three unit vectors are ar, atheta, and aphi. The unit vector ar points radially outward from the origin toward the point P. The unit vector atheta points in the direction of increasing theta (downward from the north pole along meridian lines). The unit vector aphi points in the direction of increasing phi (tangential, same as in cylindrical coordinates). All three unit vectors are position-dependent and change direction at every point in space.

The position vector in spherical coordinates is simply r = r ar. This elegant form is only possible because ar always points from the origin directly to the observation point. There are no atheta or aphi components in the position vector, which makes the spherical system especially clean for radially symmetric field problems.

Mathematical Expression

The transformation from spherical coordinates (r, theta, phi) to Cartesian coordinates (x, y, z) is: x = r sin(theta) cos(phi), y = r sin(theta) sin(phi), z = r cos(theta). The inverse transformation gives: r = sqrt(x^2 + y^2 + z^2), theta = arccos(z / r), phi = arctan(y / x).

The unit vector transformation from spherical to Cartesian is: ar = sin(theta)cos(phi) ax + sin(theta)sin(phi) ay + cos(theta) az. The unit vector atheta = cos(theta)cos(phi) ax + cos(theta)sin(phi) ay - sin(theta) az. These expressions show clearly that all three spherical unit vectors depend on both theta and phi.

The differential volume element is dV = r^2 sin(theta) dr dtheta dphi. The factor r^2 sin(theta) is called the Jacobian of the spherical transformation. It must always be included when computing total charge, total flux, or any volume integral in spherical coordinates. When integrating over a full sphere, the angular integration of sin(theta) dtheta dphi from 0 to pi in theta and 0 to 2*pi in phi gives 4*pi.

Practical Understanding

The electric field of a point charge Q located at the origin is the classic application of spherical coordinates. Using Gauss's law with a spherical Gaussian surface of radius r, the result is E = Q / (4*pi*epsilon_0*r^2) ar. This expression is valid everywhere in free space and shows that E depends only on r and points purely in the ar direction. This simplicity is only achieved in spherical coordinates.

In antenna theory, the radiation pattern of a Hertzian dipole is expressed as a function of theta in spherical coordinates. The far-field electric field varies as sin(theta) in the atheta direction. Plotting the radiation pattern requires evaluating the field at constant r as theta varies from 0 to pi.

Given:
Convert point P with spherical coordinates (r=10, theta=30 deg, phi=60 deg) to Cartesian.

Why this formula applies:
Spherical to Cartesian conversion is needed when combining
fields from multiple sources in different coordinate systems.

Formula:
x = r sin(theta) cos(phi)
y = r sin(theta) sin(phi)
z = r cos(theta)

Substitution:
theta = 30 deg: sin(30) = 0.5, cos(30) = 0.866
phi = 60 deg:   sin(60) = 0.866, cos(60) = 0.5
r = 10

Calculation:
x = 10 x 0.5 x 0.5 = 10 x 0.25 = 2.5
y = 10 x 0.5 x 0.866 = 10 x 0.433 = 4.33
z = 10 x 0.866 = 8.66

Verification:
r = sqrt(2.5^2 + 4.33^2 + 8.66^2)
r = sqrt(6.25 + 18.75 + 75.0) = sqrt(100) = 10 (correct)

Final Answer:
Cartesian coordinates of P = (2.5, 4.33, 8.66) m

Exam Tip: In GATE spherical coordinate problems, the most frequent mistake is confusing theta (polar, from z-axis, 0 to pi) with phi (azimuthal, from x-axis, 0 to 2pi). Also always include r^2 sin(theta) in the volume element. Forgetting sin(theta) or r^2 will cause integration errors.

• Spherical coordinates (r, theta, phi) are ideal for point charges, spherical conductors, and antenna radiation patterns.
• All three unit vectors ar, atheta, aphi are position-dependent. They change direction with both theta and phi.
• Position vector is simply r = r ar. The absence of theta and phi components makes Gauss's law calculations straightforward.
• Differential volume: dV = r^2 sin(theta) dr dtheta dphi. Both r^2 and sin(theta) are mandatory.
• Integrating sin(theta) dtheta over 0 to pi gives 2. Combined with dphi integrated over 0 to 2*pi gives 4*pi, the solid angle of a full sphere.

Quick Revision

• Three coordinates: r (0 to inf), theta (0 to pi, from z-axis), phi (0 to 2pi, from x-axis).
• Unit vector cycle: ar x atheta = aphi, atheta x aphi = ar, aphi x ar = atheta.
• Transformation: x = r sin(theta)cos(phi), y = r sin(theta)sin(phi), z = r cos(theta).
• Differential volume: dV = r^2 sin(theta) dr dtheta dphi. Never omit r^2 or sin(theta).
• Position vector: r = r ar. No atheta or aphi component.
• Full sphere solid angle = 4*pi steradians. Integral of sin(theta)dtheta dphi over full sphere.
• Key GATE trap: theta and phi are sometimes swapped in notation by different textbooks. In IEEE standard, theta is polar (from z-axis) and phi is azimuthal. Always verify convention used in the question.