Electric Circuits and Field
Category : Railways
Electric Circuits and Field
BASIC CONCEPTS
An electric circuit is a path in which electrons from a voltage or current source flow. The point where those electrons enter an electrical circuit is called the "source" of electrons. The point where the electrons leave an electrical circuit is called the "return" or "earth ground". The exit point is called the "return" because electrons always end up at the source when they complete the path of an electrical circuit.
The part of an electrical circuit that is between the electrons' starting point and the point where they return to the source is called an electrical circuit's "load". The load of an electrical circuit may be as simple as those that power home appliances like refrigerators, televisions, or lamps or more complicated, such as the load on the output of a hydroelectic power generating station.
At the heart of these electrical devices are made by assembly of electrical components. These component are classified in two categories i.e., active components and passive components. are components are: semiconductors, transistors, diodes and triodes, current source, voltage source. However, these devices could not function without much simpler components known as passive component these include resistors, capacitors and
Inductors
RESONANCE
Resonance in electrical circuits consisting of passive and active elements represents a particular state of the circuit when the current or voltage in the circuit is maximum or minimum with respect to the magnitude of excitation at a particular frequency, the circuit impedance being either minimum or maximum at the power factor unity.
Series Resonance
Resonance Properties of Series RLC Circuit
\[Q=\frac{1}{{{\omega }_{0}}RC}=\frac{1}{\frac{1}{\sqrt{LC}}RC}=\frac{1}{R}\sqrt{\frac{L}{C}}.\]
\[Also,\,\,Q\,=\frac{{{f}_{0}}}{Bandwidth}=\frac{\operatorname{Re}sonant\,\,frequency}{Bandwidth}\]
Parallel Resonant
Properties of Parallel Resonant LRC Circuit
Resonance Between Parallel RC and RL Circuit
Let \[{{Y}_{1}}\]= admittance of \[{{R}_{1}}C\]circuit
\[{{Y}_{2}}\]= admittance of\[{{R}_{2}}L\] circuit
Y= net admittance \[={{Y}_{1}}+{{Y}_{2}}\]
\[=\left[ \frac{{{R}_{1}}}{R_{1}^{2}+X_{C}^{2}}+\frac{{{R}_{2}}}{R_{2}^{2}+X_{L}^{2}} \right]+j\,\,\left[ \frac{{{X}_{C}}}{R_{1}^{2}+X_{C}^{2}}-\frac{{{X}_{L}}}{R_{2}^{2}+X_{L}^{2}} \right]\]
Important Point
\[{{R}_{1}}={{R}_{2}}=\sqrt{L/C}\]
\[f=\frac{1}{2\pi \sqrt{LC}}\,\,\left[ \frac{\frac{L}{C}-R_{2}^{2}}{\frac{L}{C}-R_{1}^{2}} \right]\]
Parallel Resonance of RLC Circuit
Parallel RLC circuit in this circuit, the condition for resonance circuit occurs when the susceptance part is zero.
Importance Formulae
\[{{\omega }_{1}}\frac{1}{2RC}+\sqrt{\,\,{{\left( \frac{1}{2RC} \right)}^{2}}+\frac{1}{LC}}\]
\[{{\omega }_{2}}\frac{1}{2RC}+\sqrt{\,\,{{\left( \frac{1}{2RC} \right)}^{2}}+\frac{1}{LC}}\]
Basic \[\Delta \] - Y or\[\Delta \] - A transformation
The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming me impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equation given here are valid for complex as well as real impedances.
Equations for the transformation from \[\Delta \]to Y
The general idea is to compute the impedance \[{{R}_{y}}\]at terminal node of the Y circut with impedances R' R" to adjacent nodes in the\[\Delta \]circuit by
\[{{R}_{y}}=\frac{R'R''}{\sum {{R}_{\Delta }}}\]
where\[{{R}_{\Delta }}\]are all impedance in the\[\Delta \]circuit. This yields the specific formula
\[{{R}_{1}}=\frac{{{R}_{b}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}'{{R}_{2}}=\frac{{{R}_{a}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}\]
\[{{R}_{3}}=\frac{{{R}_{a}}{{R}_{b}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}\]
Equations for the transformation from Y to \[\Delta \]
The general idea is to compute an impedance \[{{R}_{\Delta }}\] in the \[\Delta \]circuit by
\[{{R}_{\Delta }}=\frac{Rp}{{{R}_{opposite}}}\]
Where \[{{R}_{p}}={{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}\] is the sum of the products of all pairs of impedances in the Y circuit and\[{{R}_{opposite}}\]is the impedance of the node in the Y circuit which is opposite the edge with \[{{R}_{\Delta }}\]The formula for the individual edges are
Thus
\[{{R}_{a}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{1}}}\]
\[{{R}_{b}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{2}}}\]
\[{{R}_{c}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{3}}}\]
or, if using admittance instead of resistance
\[{{Y}_{a}}=\frac{{{Y}_{3}}{{Y}_{2}}}{\sum {{Y}_{Y}}},\,\,{{Y}_{b}}=\frac{{{Y}_{3}}{{Y}_{1}}}{\sum {{Y}_{Y}}},\,\,{{Y}_{c}}=\frac{{{Y}_{1}}{{Y}_{2}}}{\sum {{Y}_{Y}}}\]
Note that the general formula in Y to\[\Delta \]using admittance is similar to\[\Delta \]to Y using resistance
NETWORK THEOREMS
Superposition Theorem
This theorem finds use in solving a network where two or more sources are present and connected not in series or in parallel.
Steps for Solving a Network Using Superposition Theorem:
(i) Select a single source. Short other voltage sources and open the current sources, if internal impedances are not known. If known, replace them by their internal impedances.
(ii) Find out the current through or voltage across the required element, due to the source under consideration.
(iii) Repeat the above steps for all the sources
(iv) Add all the individual effects produced by individual sources to obtain the total current in or voltage across the element.
The venin's Theorem
Any two terminal bilateral linear d.c. circuit can be replaced by an equivalent circuit consisting of a voltage source and a series resistor
Steps to Solve a Network Using The venin's Theorem:
(i) Remove the branch impedance, through which current is required to calculate.
(ii) Calculate voltage across the open circuited terminals. This voltage is The venin's equivalent voltage\[{{V}_{th}}\]
(iii)Calculate equivalent impedance\[{{Z}_{ep}}\]as viewed through two terminals of the branch from which current is to be calculated by removing that load impedance and replacing all the dependent sources by their internal impedance.
(iv) Required current through the branch is given by \[I=\frac{{{V}_{th}}}{{{Z}_{L}}+{{Z}_{eq}}}\]
Norton's Theorem
A linear active network consisting of independent and or dependent voltage and current sources and linear bilateral network elements can be replaced by an equivalent circuit consisting of a current source in parallel with a resistance, the current source being the short circuited current across the load terminal and the resistance being the internal resistance of the source network looking through the open circuited load terminals.
This theorem is converse of The venin's theorem.
Steps to solve a network using Norton's Theorem:
(i) Short the branch through which the current is to be calculated.
(ii) Find out the current through this short circuited branch.
This current is nothing but Norton's current\[{{I}_{N}}\].
(iii) Calculate equivalent impedance\[{{Z}_{eq}}\]as viewed through two terminals of interest by removing the load impedance and making all the independent sources inactive.
(iv) Current through the branch of interest is given by,
\[I={{I}_{N}}\times \frac{{{Z}_{eq}}}{{{Z}_{eq}}+{{Z}_{L}}}\]
Maximum Power Transfer Theorem
A resistance load, being connected to a dc network, receive maximum power when the load resistance is equal to the internal resistance (The venin's equivalent resistance) of the source network as seen from the load terminals.
Steps to solve the problems related to maximum power transfer theorem:
(i) Remove the load resistance and find The venin's resistance \[({{R}_{TH}})\]of the source network looking through the open circuited load terminals.
(ii) As per maximum power transfer theorem, this \[{{R}_{TH}}\]is the load resistance of the network i.e., \[{{R}_{L}}={{R}_{TH}}\]that allows maximum power transfer.
(iii) Find the The venin's voltage (Vg) across "the open circuited load terminals.
(iv) Maximum power transfer is given by: \[\frac{V_{0}^{2}}{4{{R}_{Th}}}\]
GRAPH OF A NETWORK
When all the elements (resistance, inductance, capacitance etc.) of a network are replaced by lines with circles or dots at both ends, the configuration is then called the graph of the network. A point at which terminal or end of two or more than two elements are joined is called node (N). A line segment representing one network element or a combination of elements connected between two points is called branch (B).
Directed (or oriented) Graph: A graph is said to be directed or oriented when all nodes and branches are numbered and directions are assigned to the branches by arrows.
The graph of the network shown in Fig.
\[{{N}_{1}},\,\,{{N}_{2}},\,\,{{N}_{3}},\,\,{{N}_{4}}\,\,and\,\,{{N}_{5}}\]represent the five nodes and \[{{B}_{1}},{{B}_{2}},{{B}_{3}},{{B}_{4}}{{B}_{5}},{{B}_{6}}\,and\,{{B}_{7}}\]seven branches of the graph of a network.
Oriented/directed graph
Definitions Related to Graph of a Network
Degree of a Node: It is the number of branches incident to it.
Loop: It is the closed contour selected in the graph.
Path: An ordered sequence of branches traversing from one node to another.
Circuit: In the network graph, it is a set of branches such that exactly two branches are incident to each of the nodes in the set. A circuit subgraph is always connected.
Subgraph: A graph is said to be the subgraph of a graph if every node and branch of subgraph is the node and branch respectively of the graph.
Connected and Non-connected graph: A graph where at least one path along branches between every pair of nodes of a graph exists.
Connected graph Non-connected graph
Tree and Co-tree: Tree is an interconnected open set of branches which include all the nodes of the given graph. In a tree of a graph there cannot be any closed loop. A branch of tree is known as twig. Those branches of a graph which are not included a tree are called co-tree. The branches of a co-tree are called links or chords.
Total no. of links L=B-(N- 1) =B-N+ 1
where B = total no. of branches
N-1 = total no. of tree branches
Cut-set: It is that set of elements or branches of a graph that separates two main parts of a network. If any branch of the cut-set is not removed, the network remains connected. The term cut-set is derived from the property designated by the way by which the network can be divided into two parts.
A cut set is shown on a graph by a dashed line, where the dashed line passes through the branches denning the cut-set. A graph should have at least one cut-set through there may be more than one cut-set in any graph.
Properties of a Tree
(i) It consists of all the nodes of the graph.
(ii) If the graph has N no. of nodes, the tree will have (N-(iii) branches.
(iii) There will be no closed path in the tree.
(iv) There can be many possible different trees for a given graph depending on the number of nodes and branches.
Number of independent KCL equations \[=N-1\]
Rank of a graph.
If there exists N number of nodes, then rank R of a graph is given by the relation
\[R=\left( N-1 \right)\]
No. of fundamental cut-sets = no. of twigs \[=\left( N-1 \right)\]where N = no. of nodes of a graph
Number of independent node equations (n) = J (no. of junctions) \[-1.\]
Number of independent mesh equations (m) = b (no. of branches) \[-(j-1)\]
TWO-PORT NETWORK
In the two port network, there are four variables - two voltages and two currents. We use \[{{V}_{1}}\] and \[{{I}_{1}}\] as variables at the input and \[{{V}_{2}}\]and\[{{I}_{2}}\] as variables at the output as shown in the figure.
Z-parameters (impedance parameter or Open circuit parameter)
Impedance parameter or Z-parameter (open circuit impedance parameters)––
Here, \[{{V}_{1}}\] and \[{{V}_{2}}\] are expressed in terms of \[{{I}_{1}}\]and \[{{I}_{2}}\]
\[{{V}_{1}}={{Z}_{11}}{{I}_{1}}+{{Z}_{12}}{{I}_{2}}\]
\[{{V}_{2}}={{Z}_{21}}{{I}_{1}}+{{Z}_{12}}{{I}_{2}}\]
\[{{Z}_{11}}\]= impedance when output is open circuited
\[=\frac{{{V}_{1}}}{{{l}_{1}}}\,\,when\,\,{{I}_{2}}=0\]
\[{{Z}_{22}}\]= impedance when input is open circuited
\[=\frac{{{V}_{2}}}{{{l}_{2}}}\,\,when\,\,{{I}_{1}}=0\]
\[{{Z}_{12}}\]= forward impedance when input is open circuited
\[=\frac{{{V}_{1}}}{{{l}_{1}}}\,\,when\,\,{{I}_{1}}=0\]
\[{{Z}_{21}}\]= reverse impedance when output is open circuited
\[=\frac{{{V}_{2}}}{{{l}_{1}}}\,\,when\,\,{{I}_{2}}=0\]
Admittance Parameter or Y-Parameter (Short Circuit Parameter)
Here \[{{I}_{1}}\]and \[{{I}_{2}}\]are expressed in terms of\[{{V}_{1}}\] and \[{{V}_{2}}\]
\[{{I}_{1}}={{Y}_{11}}{{V}_{1}}+{{Y}_{12}}{{V}_{2}}\]
\[{{I}_{2}}={{Y}_{21}}{{V}_{1}}+{{Y}_{22}}{{V}_{2}}\]
\[{{Y}_{11}}\]= admittance when output is short circuited \[=\frac{{{I}_{1}}}{{{V}_{1}}}\]when \[{{V}_{2}}=0\]
\[{{Y}_{22}}\]= admittance when input is short circuited \[=\frac{{{I}_{2}}}{{{V}_{2}}}\]when \[{{V}_{1}}=0\]
\[{{Y}_{12}}\]=forward admittance when input is short circuited \[=\frac{{{I}_{1}}}{{{V}_{2}}}\]when\[{{V}_{1}}=0\]
\[{{Y}_{21}}\]=reverse admittance when output is short circuits \[=\frac{{{I}_{2}}}{{{V}_{1}}}\,\,when\,\,{{V}_{2}}=0\]
Hybrid Parameter or h-Parameter
\[{{V}_{1}}={{h}_{11}}{{I}_{1}}+{{h}_{12}}{{V}_{2}}\]
\[{{I}_{2}}={{h}_{21}}{{I}_{1}}+{{h}_{22}}{{V}_{2}}\]
\[{{h}_{11}}=\]short circuited input impedance\[=\frac{{{V}_{1}}}{{{I}_{1}}}when\,\,{{V}_{2}}=0\]
\[{{h}_{22}}\]\[=\]open circuited output admittance \[=\frac{{{I}_{2}}}{{{V}_{2}}}when\,\,{{I}_{1}}=0\]
\[{{h}_{12}}\]open circuit reverse voltage gain \[=\frac{{{V}_{1}}}{{{V}_{2}}}when\,\,{{I}_{1}}=0\]
\[{{h}_{21}}=\]short circuit current gain \[=\frac{{{I}_{2}}}{{{I}_{1}}}when\,\,{{V}_{2}}=0\]
ABCD Parameters (Transmission Parameter)
ABCD parameters are widely used in analysis of power transmission engineering where they are termed as "Generalised circuit Parameters" ABCD parameters are also called as transmission Parameters".
Representation of input and output voltages and currents in two-port network for ABCD parameter-representation
\[{{V}_{1}}=A{{V}_{2}}+B\,\,(-{{I}_{2}})\]
\[{{I}_{1}}=C{{V}_{2}}+D\,\,(-{{I}_{2}})\]
\[A={{\left. \frac{{{V}_{1}}}{{{V}_{2}}} \right|}_{{{I}_{2}}=0}}\]
\[C={{\left. \frac{{{I}_{1}}}{{{V}_{2}}} \right|}_{{{I}_{2}}=0}}\]
‘A’ called is reverse voltage ratio and does not have any unit. 'C' known as transfer admittance and has the unit mho.
\[B={{\left. \frac{{{V}_{1}}}{-{{I}_{2}}} \right|}_{{{V}_{2}}=0}}\]
\[D={{\left. \frac{{{I}_{1}}}{-{{I}_{2}}} \right|}_{{{V}_{2}}=0}}\]
‘D’ being a ratio of two currents, it is called reverse current ratio; It is an unitless quantity while 'B' is expressed in ohm and is termed as transfer impedance.
Table: Condition of Reciprocity and Symmetry in Terms of Various parameters
|
parameter |
Condition for reciprocity |
Condition for symmetry |
|
Z |
\[{{Z}_{12}}={{Z}_{21}}\] |
\[{{Z}_{11}}={{Z}_{22}}\] |
|
Y |
\[{{Y}_{12}}={{Y}_{21}}\] |
\[{{Y}_{11}}={{Y}_{21}}\] |
|
H |
\[{{h}_{12}}={{h}_{21}}\] |
\[\Delta h=1\] |
|
ABCD |
AD-BC=1 |
A=D |
THREE – PHASE CIRCUITS
Star Connection or Y – Connection
Line voltage, \[{{V}_{L}}=\sqrt{3}\,\,({{V}_{Ph}})\] where \[{{V}_{Ph}}\]is the phase voltage
Line current, \[{{I}_{L}}={{I}_{Ph}};\] where \[{{I}_{Ph}}\]is the phase current.
Total active power,\[P=3\times {{V}_{Ph}}\times {{I}_{Ph}}\times \cos \Phi ;\]where \[\Phi \] is the angle between phase voltage and phase current.
Total active power,\[P=\sqrt{3}\times {{V}_{L}}\times {{I}_{L}}\times \cos \Phi \,\,;\] where \[\Phi \] is the angle between phase voltage and phase current.
Delta Connection or D-Connection
Line voltage, \[{{V}_{L}}={{V}_{Ph}}\] where \[{{V}_{Ph}}\] is the phase voltage.
Line current, \[{{I}_{L}}=\sqrt{3}\times {{I}_{Ph}}\,\,;\] where\[{{I}_{Ph}}\]is the phase current.
Total active power, where \[P=3\times {{V}_{Ph}}\times {{I}_{Ph}}\times \cos \Phi \] where\[\Phi \]is the angle between phase voltage and phase current. Total active power,\[P=\sqrt{3}\times {{V}_{L}}\times {{I}_{L}}\times \cos \Phi \,\,;\] where \[\Phi \] is the angle between phase voltage and phase current.
Kirchhoff’s Current Law (KCL)
This law states that at any node, the algebraic sum of all branch currents leaving and entering a node at any instant of time is zero. In other words, at any instant, the algebraic sum of the magnitude of the different currents entering a node is equal to the algebraic sum of the magnitude of the currents leaving that node.
Applying KCL in fig. (a), it can be said that
\[{{I}_{3}}+{{I}_{6}}={{I}_{1}}+{{I}_{2}}+{{I}_{4}}+{{I}_{5}}\]
Thus, current entering a Bode = current leaving a node
Kirchhoff’s Voltage Law (KVL)
In any closed loop of a network, the algebraic sum of all branch voltages is zero.
Applying KVL in fig, (b), we can say that
\[{{V}_{1}}+{{V}_{2}}={{I}_{1}}{{R}_{1}}+{{I}_{1}}{{R}_{2}}+{{I}_{1}}{{R}_{3}}\]
\[{{V}_{1}}+{{V}_{2}}={{I}_{1}}{{R}_{1}}-{{I}_{1}}{{R}_{2}}-{{I}_{1}}{{R}_{3}}=0\]
Nodal Analysis
(i) Label all nodes in the circuit.
(ii) Arbitrarily select any node as reference.
(iii) Define a voltage variable from every remaining node to the reference. These voltage variables must be defined as voltage rises with respect to the reference node.
(iv) Write a K-CL equation for every node except the reference.
(v) Solve the resulting system of equations.
(vi)Number of independent node equations (n) = J (no. of junctions)\[-1\].
Writing KCL equations for the network shown in figure below,
Node 1: \[{{I}_{1}}-\frac{{{V}_{1}}}{{{R}_{1}}}-\frac{({{V}_{1}}-{{V}_{2}})}{{{R}_{2}}}=0\] ...(i)
Node 2: \[\frac{({{V}_{1}}-{{V}_{2}})}{{{R}_{2}}}-\frac{{{V}_{2}}}{{{R}_{3}}}+{{I}_{2}}=0\] ...(ii)
Rearranging equation (i) and (ii), we get
\[{{V}_{1}}\,\,\left[ \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}} \right]-{{V}_{2}}\,\,\left[ \frac{1}{{{R}_{2}}} \right]={{I}_{1}}\,\,and\]
\[{{V}_{1}}\,\,\left[ \frac{-1}{{{R}_{2}}} \right]+{{V}_{2}}\,\,\left[ \frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}} \right]={{I}_{2}}\]
These equations can be solved simultaneously to get unknown node voltages from which any branch current can be calculated.
Mesh Analysis
Mesh is a loop that does not contain an inner loop.
(i) Count the number of "window panes" in the circuit.
(ii) Assign a mesh current to each window pane.
(iii) Write a KVL equation for every mesh whose current I unknown.
(iv) Solve the resulting equations.
(v)Number of independent mesh equations (m)=b (no. of branches)\[-\left( j-1 \right)\]
Thus, loop equations for the network shown in figure below are:
For loop a – b – e – f – a,
\[-{{I}_{1}}{{R}_{1}}-({{I}_{1}}-{{I}_{2}}){{R}_{3}}+{{V}_{1}}=0\]
For loop b – c – d – e - b,
\[{{R}_{2}}{{I}_{2}}+({{I}_{2}}-{{I}_{1}})\,\,{{R}_{3}}+{{V}_{2}}=0\]
ELECTRIC FIELD
An electric field is a vector field that associates to each point in space the Coulomb force that would be experienced per unit electric charge, by an infinitesimal test charge at that point. Electric fields converge and diverge at electric charges and can be induced by time-varying magnetic fields.
Electric field due to an infinite long straight charged line:
Consider one example of an infinite long straight charged line having uniform linear charge density (k) and a point P located at a perpendicular distance r from the linear charge distribution The electric field intensity at any point P and all the other points like P situated perpendicular distance from the line will be equal as the line is of infinite length (L).
As per Gauss' law, the electric field intensity at point P on an infinitely long straight charged line is:
\[E=\frac{\lambda }{2\pi {{\varepsilon }_{0}}}\times \frac{1}{r}\times \hat{r}\]
Here we have
\[\lambda \]= linear charge density
\[{{\varepsilon }_{0}}\]= electrical permittivity of free space
r= radius
\[\hat{r}\]= unit vector in the direction of radius.
GAUSS THEOREM
Gauss theorem states that the surface integral of the normal component of the electric intensity, E over a closed surface is always equal to\[{}^{1}/{}_{{{\varepsilon }_{0}}}\] times the total charge (Q) inside it.
\[E\times area=\frac{Q}{{{\varepsilon }_{0}}}\]
Consider one example of an infinite long straight charge line having uniform linear charge density \[(\lambda )\] and a point P located at a perpendicular distance r from the linear charge distribution. The electric field intensity at any point P and all the other points like P situated perpendicular distance from the line will be equal as the line is of infinite length (L). As per Gauss' law, the electric field intensity at point P on an infinitely long straight charged line is:
\[E=\frac{\lambda }{2\pi {{\varepsilon }_{0}}}\times \frac{1}{r}\times \hat{r}(\lambda )\]
Here we have
\[\lambda \]= linear charge density
\[{{\varepsilon }_{0}}\]= electrical permittivity of free space
r= radius
\[\hat{r}\]= unit vector in the direction of radius.
\[E=\frac{\sigma }{2{{\varepsilon }_{0}}}\]
Here we have
\[\sigma \]= Surface charge density
\[{{\varepsilon }_{0}}\]= electrical permittivity of free space
It shows that the electric field intensity at any point on the plane sheet is not depend on the distance of the point from the plane.
Consider a spherical shell having surface charge density o and radius R. The electric field resulting from such a spherical shell is radial and hence electric field intensity is calculated for a point lying inside and outside the spherical shell.
(a) Point lying inside the shell: Here point is lying inside the shell and having radius r smaller then the spherical shell radius R. So, as per the Gauss' law, the electric field intensity is zero due to charge enclosed by such a surface is zero as the radius is concentric with the shell.
(b) Point lying outside the shell: Here point is lying outside the shell and having radius r greater than the spherical shell radius R. So, as per the Gauss law, the electric field intensity is,
\[E=\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{1}{{{r}^{2}}}\]
It shows that for a point outside the sphere, the entire charge of the sphere can be treated as concentrated at its centre.
(a) Point lying inside the shell: Here point is lying inside the sphere and the spherical Gaussian surface of radius r < R, concentric with the sphere. Using Gauss' law, the electric field intensity is
\[E=\frac{pr}{3{{\varepsilon }_{0}}}\]
(b) Point lying outside the shell: Here point is lying outside the sphere and the spherical Gaussian surface of radius r > R, coincide with the each other. Using
Gauss law, the electric field intensity is
\[E=\frac{{{R}^{3}}\rho }{3{{r}^{2}}{{\varepsilon }_{0}}}\]
AMPERE AND BIOT-SAVARTS LAW
According to the Biot-Savart Law,
\[dB=\frac{{{\mu }_{0}}}{4\pi }.\frac{Idl\sin \theta }{{{r}^{2}}}Wb/{{m}^{2}}\]
Where
dB= magnetic field strength
I = current
dl=length of current element
r=distance between the observation point and the line element
\[\theta \]=angle between the observation point and the line element
INDUCTANCE
Inductance is the property or the behaviour of a coil of wire because of which it resists any change of electric current flowing through it.
\[L=N\frac{d\phi }{di}\]
MAGNETIC FIELD
Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to many practical applications. Magnetic field sources are essentially dipolar in nature, having a north and south magnetic pole. The SI unit for magnetic field is the Tesia, which can be seen from the magnetic part of the Lorentz force law \[{{F}_{magnetic}}=qvB\]to be composed of (Newton\[\times \]second)/ (Coulomb\[\times \]meter). A smaller magnetic field unit is the Gauss (1 Tesia = 10,000 Gauss).
BIOT-SAVART LAW
The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner. Coulomb's law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing.
Magnetic field of a current element \[d\overrightarrow{B}=\frac{{{\mu }_{0}}Id\overrightarrow{L}x\overrightarrow{1r}}{4\pi {{r}^{2}}}\]
where, \[d\overrightarrow{B}\]= magnetic field strength.
\[d\overrightarrow{L}\]= infinitesmal length of conductor carrying electric current I
\[\overrightarrow{{{1}_{r}}}\]= unit vector to specify the direction of the vector distance r from the current to the field point.
See the magnetic field sketched for the straight wire to see the geometry of the magnetic field of a current.
AMPERE'S LAW
The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.
In the electric case, the relation of field to source is quantified in
Gauss's Law which is a very powerful tool for calculating electric fields.
Electromagnetism
It is a branch of physics which involves the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields, such as electric fields, magnetic fields, and light. The electromagnetic force is one of the four fundamental interactions (commonly called forces) in nature.
The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation.
Magnetic effect of electric current carrying conductor:
Right hand thumb rule.
Magnetic Field of Current Loop
Examining the direction of the magnetic field produced by a current-carrying segment of wire shows that all parts of the loop contribute magnetic field in the same direction inside the loop.
Electric current in a circular loop creates a magnetic field which is more concentrated in the center of the loop than outside the loop. Stacking multiple loops concentrates the field even more into what is called a solenoid.
MAGNETIC MATERIALS
Materials respond differently to the force of a magnetic field. There are three main classifications of magnetic materials. A magnet will strongly attract ferromagnetic materials, weakly attract paramagnetic materials and weakly repel diamagnetic materials.
The orientation of the spin of the electrons in an atom, the orientation of the atoms in a molecule or alloy, and the ability of domains of atoms or molecules to line up are the factors that determine how a material responds to a magnetic field. Ferromagnetic materials have the most magnetic uses. Diamagnetic materials are mainly used in magnetic levitation.
Ferromagnetic materials
Ferromagnetic materials are strongly attracted by a magnetic force. The elements iron (Fe), nickel (Ni), cobalt (Co) and gadolinium (Gd) are such materials.
The reasons these metals are strongly attracted are because their individual atoms have a slightly higher degree of magnetism due to their configuration of electrons, their atoms readily line up in the same magnetic direction and the magnetic domains or groups of atoms line up more readily.
Paramagnetic materials
Paramagnetic materials are metals that are weakly attracted to magnets. Aluminum and copper are such metals. These materials can become very weak magnets, but their attractive force can only be measured with sensitive instruments.
Temperature can affect the magnetic properties of a material. Paramagnetic materials like aluminum, uranium and platinum become more magnetic when they are very cold.
The force of a ferromagnetic magnet is about a million times that of a magnet made with a paramagnetic material. Since the attractive force is so small, paramagnetic materials are typically considered nonmagnetic.
Diamagnetic materials
Certain materials are diamagnetic, which means that when they are exposed to a strong magnetic field, they induce a weak magnetic field in the opposite direction. In other words, they weakly repel a strong magnet. Some have been used in simple levitation demonstrations.
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