Vector Algebra 2

In this post we'll lern Vector algebra in component form.
Component of any vector is the projection of that vector along the three coordinate axis X, Y, Z.

VECTOR ADDITION
In component form addition of two vectors is
C = (Ax+ Bx)i + (Ay+ By)j + (Ay+ By)k
where,
A = (Ax, Ay, Az) and B = (Bx, By, Bz)
Thus in component form resultant vector C becomes,
Cx = Ax+ Bx
Cy = Ay+ By
Cz = Az+ Bz

SUBTRACTION OF TWO VECTORS
In component form subtraction of two vectors is
D = (Ax- Bx)i + (Ay- By)j + (Ay- By)k
where,
A = (Ax, Ay, Az) and B = (Bx, By, Bz)
Thus in component form resultant vector D becomes,
Dx = Ax - Bx
Dy = Ay- By
Dz = Az- Bz

NOTE:- Two vectors add or subtract like components.

DOT PRODUCT OF TWO VECTORS
A.B = (Axi + Ayj + Azk) . (Bxi + Byj + Bzk)
= AxBx + AyBy + AzBz.
Thus for calculating the dot product of two vectors, first multiply like components, and then add.

CROSS PRODUCT OF TWO VECTORS

A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)
= (AyBz - AzBy)i + (AzBx - AxBz)j + ( AxBy - AyBx)k.

Cross product of two vectors is itself a vector.
To calculate the cross product, form the determinantwhose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

VECTOR TRIPPLE PRODUCT

Vector product of two vectors can be made to undergo dot or cross product with any third vector.

(a) Scalar tripple product:-
For three vectors A, B, and C, their scalar triple product is defined as
A . (B x C) = B . (C x A) = C . (A x B)
obtained in cyclic permutation. If A = (Ax, Ay, Az) , B = (Bx, By, Bz) , and C = (Cx, Cy, Cz) then A . (B x C) is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the 3 x 3 matrix formed by A, B, and C.

(b) Vector Triple Product:-
For vectors A, B, and C, we define the vector tiple product as
A x (B x C) = B(A . C) - C(A - B)
Note that
(A . B)C ≠ A(B . C)
but
(A . B)C = C(A . B).

Electric resistance and Resistivity

(A) Resistivity

  • In the previous post we derived that current density is
    j = nqvd
    where vd is the drift velocity.

  • Current density in general depend on electric field and for metals current density is nearly proportional to the electric field. (Results can be derived using theory of metallic conduction.)

  • Thus for metals ratio of E and j is constant and for a particular material its resistivity ρ is defined as the ratio of magnitude of electric field to current density,
    ρ = E/j
    This relationship is known as Ohm's law discovered by german physicist Georg Simon Ohm (1787-1854) in 1826.

  • Greater would be the resistivity of a given material greater field would be required to establish a given current density in the material or we can say that smaller would be the current density for a given field.

  • Unit of resistivity is Ωm (ohm. meter).

  • Materials having zero resistivity are known as perfect conductors and those having infinite resistivities are known as perfect insulators. Real materials lie between these two limits.

  • Metals and alloys are materials having lowest resistivities and are good conductors of electricity.

  • Insulators have resistivities many times (of the order of 1022) greater then that of metals.

  • Reciprocal of resistivity is conductivity. Unit of conductivity is (Ωm)-1.

  • Metals or good conductors of electricity have conductivity greater than that of insulators.

  • Semiconductors are those materials which have resistivities intermediate between those of metals and insulators.


    (B) Resistivity and temperature

  • Resistivity of a conductor depends on a number of factors and temperature of the metal is one such factor. As the temperature of the conductor is increased its resistivity also increases.

  • For small variations in temperature resistivity of materials is given by the relation
    ρ(T) = ρ(T0)[ 1 +α(T-T0)]
    where, ρ(T) and ρ(T0) are resistivities at temperature T and T0 respectively and α is constant for a given material which also depends on temperature to a small extent. This constant α is known as temperature coefficent of resistivity.

    (C) Resistance

  • We already know that for a conducror relation between electric field E and current density is given as
    E = ρj
    where ρ is a constant independent of E.

  • When we study electric circuits we are more interested in the total current in a conductor rather then current density j and more interested in knowing the potential difference between the ends of the conductor than in Electric field becaude current and potential difference are easier to measure then j and E.

  • Consider a conducting wire of length l and uniform crossectional area A. If V is the potential difference between both the ends of the wire then electric field inside the conductor would be
    E = V/l
    If i is the current flowing inside the wire then current density is given by
    j = i/A
    putting these values in Ohm's law ρ = E/j we get
    V = ρi (l/A)
    or , V=Ri
    where, R=ρ(l/A)
    which is known as resistance of a given conductor.

  • Unit of resistance is ohm or volt per ampere.

  • Thus how much current will flow in a wire not only depends on the potential difference between two ends of the wire but also on the resistance offered by the conductor to the flow of electric charge.

  • From the above discussion we can easily conclude that The resistance of a wire depends both on the thickness and length of the wire and also on its resistivity.

  • Thick wires have less resistance then thin ones and longer wires have more resistance then shorter ones.

  • Since the resistivity of a marerial varies with temperature, the resistance of any particular conductor also varies with temperature. For temperature ranges that are not too great, this variation is approximately a linear relationship, analogous to the one we learned for resistivity
    R(T) = R(T0)[1 + α(T - T0)]
    In this equation. R (T) is the resistance at temperature T and R(T0) is the resistance at temperature T0. The temperature coefficient of resistance α is the same constant that appears in case of resistivity.

    In the next post we'll do some worked examples related to this topic
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