Engineering Physics ICT 111

Unit I

Scalars, Vectors and Tensors: Physical Quantities, Scalars and Vectors, Vector Algebra, Vector Addition, Vector Subtraction, Triangular Law, Parallelogram Law and Tensors.

Physical Quantities: Physical quantities are quantities used to describe the physical world around us. A physical quantity possesses a certain magnitude and is measured in specific units. Depending on the nature of the quantity, direction may also be associated with it.

Scalars: Scalar quantities are physical quantities that possess magnitude and a unit in which they are measured, but they do not require a direction for their description. Examples of scalar quantities include mass, temperature, volume, time, distance, work, power, pressure, density, frequency, entropy, specific heat capacity, refractive index and luminous intensity.

Scalar Fields: A field represents a spread or distribution of a physical quantity over a region. The study material introduces the example of the Earth having a gravitational field, which is broadly categorized as a potential field.

Vectors: Vector quantities are physical quantities that possess magnitude, a unit in which they are measured and a direction. Examples include displacement, velocity, force, acceleration and momentum.

Vector Algebra: Vector algebra deals with mathematical operations involving vectors. The study material introduces vector addition and vector subtraction. Vectors can be added using the head-to-tail method, where the tail of one vector is placed at the head of the preceding vector.

Vector Addition: Two or more vectors can be added using the head-to-tail method. The resultant vector is drawn from the tail of the first vector to the head of the final vector. The resultant can be represented as:

R⃗ = A⃗ + B⃗ + … + D⃗ + … + N⃗

Triangular Law: In the triangular law of vector addition, one vector is placed along a plane and the tail of the second vector is placed at the head of the first vector. Additional vectors are placed successively in the same head-to-tail manner. The resultant vector is drawn from the tail of the first vector to the head of the final vector.

Parallelogram Law: In the parallelogram law, the two vectors are placed with their tails at the same initial point. Lines parallel to each of the two vectors are drawn to complete the parallelogram. The diagonal drawn from the common origin of the two vectors represents the resultant vector.

Steps of Parallelogram Law:

1. Draw the tails of vectors A⃗ and B⃗ from the same initial point.
2. Draw lines parallel to each of the two vectors to complete the parallelogram.
3. Draw the diagonal from the common point of origin of the two vectors to represent the resultant vector.

Magnitude of Resultant Vector: The magnitude of the resultant vector obtained by adding two vectors A and B is given by:

R = |A + B| = √(A² + B² + 2AB cos θ)

Direction of Resultant Vector: The direction of the resultant vector can be determined using the tangent relation involving the magnitudes of the vectors and the angle between them.

Vector Subtraction: Vector subtraction is included as a part of vector algebra. The study material introduces vector subtraction after discussing vector addition and the parallelogram law. The PDF does not provide a detailed step-by-step explanation of vector subtraction.

Tensors: Tensors generalise both scalar and vector quantities in physics by relating multiple directions and planes together. A tensor is usually represented as an array of numbers or functions that transform predictably when the coordinate system is changed.

Rank of a Tensor: The rank of a tensor defines the number of directions associated with the physical quantity represented by the tensor.

Rank (0) Tensor: A rank (0), or zero-rank, tensor requires zero directions and is represented by a single number. Examples include temperature and mass.

Rank (1) Tensor: A rank (1), or one-rank, tensor requires one direction and represents a one-dimensional quantity. Examples include velocity and force.

Rank (2) Tensor: A rank (2), or two-rank, tensor requires two directions and represents quantities associated with two dimensions. Examples include inertia, stress and moment of inertia. The study material also notes that this can be extended to a 3 × 3 array or a three-dimensional space.

Applications of Tensors: Tensors are applied across physics to handle multi-axis properties such as stress, space-time curvature and electromagnetic fields. They can also map one set of directional quantities to another, such as relating an electric field to a polarization response in anisotropic crystals.

[As per the provided study material]