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Vectors in EMFT | GATE ECE

Vectors form the backbone of Electromagnetic Field Theory (EMFT). Almost every physical quantity like electric field, magnetic field, and flux density is represented using vectors.

Understanding vectors is not just mathematical — it is essential to visualize how fields behave in space.

Keywords: vectors EMFT, gradient divergence curl, coordinate systems EMFT, vector algebra GATE, del operator, field theory basics

What is a Vector?

A vector is a quantity that has both magnitude and direction.

Examples: Electric Field (E), Magnetic Field (H), Displacement (D)

$$ \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} $$

Key Idea:

  • Magnitude tells "how much"
  • Direction tells "where"
Click to zoom-in the images
Vector representation
Vector showing magnitude and direction,
Here, x=3 and y=2

Types of Vectors

Type Meaning Example
Unit Vector Magnitude = 1 $\hat{i}, \hat{j}, \hat{k}$
Position Vector Vector Form $\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}$
Zero Vector Magnitude = 0 $\vec{0}$

Vector Operations

Dot Product (Scalar)

$$ \vec{A} \cdot \vec{B} = AB \cos\theta $$

  • Result is scalar
  • Used in work, projection

Cross Product (Vector)

$$ \vec{A} \times \vec{B} = AB \sin\theta \ \hat{n} $$

  • Result is vector
  • Direction by right-hand rule
Dot and cross product
Dot vs Cross Product visualization

Differential Operators (CORE PART)

Gradient

$$ \nabla V $$

Direction of maximum increase

Divergence

$$ \nabla \cdot \vec{A} $$

Measures source or sink

Curl

$$ \nabla \times \vec{A} $$

Measures rotation

Gradient divergence curl
Physical meaning of Gradient, Divergence and Curl

Vector Projection (Very Important)

Vector projection helps us find how much of one vector lies along another vector.

Projection of $\vec{A}$ on $\vec{B}$:

$$ \text{Projection} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} $$

Vector Form:

$$ \text{Proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} $$

Key Understanding:

  • Gives component of $\vec{A}$ along $\vec{B}$
  • Always along direction of $\vec{B}$
  • Uses dot product (cosine relation)

Physical Meaning:

  • Work done = projection of force along displacement
  • Only parallel component contributes
Vector projection diagram showing A projected onto B
Projection of vector A onto vector B
Click to zoom-in

Physical Meaning (Where Students Struggle)

Operator Meaning Visualization
Gradient Maximum increase Hill slope
Divergence Outflow / Source Water spreading
Curl Rotation Whirlpool

GATE Level Insights

  • Divergence ≠ 0 → Charge present

    • If $\nabla \cdot \vec{D} > 0$ → positive charge (source)
    If $\nabla \cdot \vec{D} < 0$ → negative charge (sink)

  • Curl ≠ 0 → Rotational field
  • Curl = 0 → Conservative field

Common Mistakes

  • Mixing coordinate systems
  • Wrong unit vectors
  • Confusing divergence and curl
  • Forgetting determinant in cross product

Quick Revision Trick

  • Dot → Scalar → Cos
  • Cross → Vector → Sin
  • Grad → Increase
  • Div → Outflow
  • Curl → Rotation

Mini Practice (GATE Level)

Q: If $\vec{A} = x\hat{i} + y\hat{j} + z\hat{k}$, find divergence.

Solution:

$$ \nabla \cdot \vec{A} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3 $$

Discussion / Comments


Next: Coordinate Systems (Cartesian, Cylindrical & Spherical)