Cartesian Coordinate System

The Cartesian coordinate system is the most fundamental framework used in electromagnetics to describe the position of points, the direction of fields, and the geometry of physical structures. Every electromagnetic quantity, from electric field vectors to current distributions, must be expressed in a coordinate system, and the Cartesian system forms the starting point before more advanced systems like cylindrical and spherical are introduced.

For GATE electromagnetics and university courses, a solid understanding of the Cartesian system, including unit vectors, position vectors, and differential elements, is essential before attempting field calculations or applying Maxwell's equations.

Core Concept Explanation

The Cartesian system uses three mutually perpendicular axes, x, y, and z, to uniquely identify every point in three-dimensional space. The three unit vectors ax, ay, and az point in the positive x, y, and z directions respectively. Each unit vector has magnitude 1 and is dimensionless. They form a right-handed coordinate system, meaning that ax cross ay equals az.

A position vector r locates a point P in space relative to the origin. If P has coordinates (x, y, z), the position vector is written as r = x ax + y ay + z az, and its magnitude is |r| = sqrt(x^2 + y^2 + z^2). This vector completely describes where the point is and in what direction it lies from the origin.

The unit vectors in Cartesian coordinates are constant everywhere in space. Their direction and magnitude do not change with position. This is a unique property that does not hold in cylindrical or spherical systems, where unit vectors change direction depending on where the observation point is located.

A general vector field A in Cartesian coordinates is written as A = Ax ax + Ay ay + Az az, where Ax, Ay, Az are the scalar components of A along each axis. These components can themselves be functions of x, y, and z in a non-uniform field.

Mathematical Expression

The distance between two points P1(x1, y1, z1) and P2(x2, y2, z2) is computed as the magnitude of the difference vector. The distance vector from P1 to P2 is r12 = (x2 - x1)ax + (y2 - y1)ay + (z2 - z1)az, and the scalar distance is |r12| = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).

The differential volume element in Cartesian coordinates is dV = dx dy dz. The differential surface elements are dSx = dy dz ax, dSy = dx dz ay, and dSz = dx dy az, depending on which face of the differential cube is being considered. The differential line element is dl = dx ax + dy ay + dz az.

The dot product of two vectors A and B is A.B = Ax Bx + Ay By + Az Bz. The cross product is computed using the 3x3 determinant with ax, ay, az in the first row. These operations are used constantly in computing electric flux, work done by a field, and torque in electromagnetic problems.

Practical Understanding

In electromagnetic field problems, the Cartesian system is most convenient when the geometry of the problem is rectangular. For example, parallel plate capacitors, rectangular waveguides, and planar transmission lines are naturally described in Cartesian coordinates. The field components align cleanly with the axes, making integration and differentiation straightforward.

When computing the electric field due to a point charge using Coulomb's law, the unit distance vector aR = r12 / |r12| is needed. This is constructed by dividing the distance vector by its own magnitude. In Cartesian coordinates this calculation is direct and systematic.

Given:
Point P1 = (1, 2, 3) m and P2 = (4, 6, 3) m
Find: distance vector r12, its magnitude, and unit vector

Why this formula applies:
Distance vector and unit vector are fundamental for Coulomb's law
and field direction calculations in electromagnetics.

Formula:
r12 = (x2 - x1)ax + (y2 - y1)ay + (z2 - z1)az
|r12| = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
ar12 = r12 / |r12|

Substitution:
r12 = (4-1)ax + (6-2)ay + (3-3)az
r12 = 3ax + 4ay + 0az

Calculation:
|r12| = sqrt(3^2 + 4^2 + 0^2)
|r12| = sqrt(9 + 16) = sqrt(25) = 5 m

ar12 = (3ax + 4ay) / 5
ar12 = 0.6 ax + 0.8 ay

Final Answer:
Distance vector r12 = 3ax + 4ay m
Magnitude |r12| = 5 m
Unit vector ar12 = 0.6 ax + 0.8 ay (dimensionless, magnitude = 1)

Exam Tip: In GATE, always verify that your unit vector has magnitude exactly 1 by squaring and summing components. A common mistake is forgetting to divide the distance vector by its magnitude when computing the unit vector for Coulomb's law or field direction.

• Cartesian unit vectors ax, ay, az are constant in direction everywhere in space, unlike cylindrical and spherical unit vectors.
• The right-hand rule gives the cross product direction: ax x ay = az, ay x az = ax, az x ax = ay. Reversing the order negates the result.
• Position vector r = x ax + y ay + z az. Its magnitude gives the distance from origin to point P.
• Distance vector between two points r12 = r2 - r1. Unit vector ar12 = r12 / |r12|.
• Differential volume dV = dx dy dz is used in volume charge density integration to find total charge enclosed in a region.

Quick Revision

• Three orthogonal axes: x, y, z. Unit vectors ax, ay, az with magnitude 1. Right-handed system.
• Position vector: r = x ax + y ay + z az. Magnitude |r| = sqrt(x^2 + y^2 + z^2).
• Dot product: A.B = AxBx + AyBy + AzBz. Scalar result. Used in work and flux calculations.
• Cross product: 3x3 determinant. Vector result. Used in force and torque calculations.
• Cyclic cross products: ax x ay = az, ay x az = ax, az x ax = ay. Anti-cyclic gives negative.
• Differential elements: dl = dx ax + dy ay + dz az, dV = dx dy dz. Essential for integration.
• Key GATE trap: unit vectors in Cartesian do not change with position. In cylindrical and spherical they do change, which is why transformation rules must be applied carefully.