Electric flux

The field around a charged conductor

A conductor is in electrostatic equilibrium when the charge distribution (the way the charge is distributed over the conductor) is fixed. Basically, when you charge a conductor the charge spreads itself out. At equilibrium, the charge and electric field follow these guidelines:

• the excess charge lies only at the surface of the conductor
• the electric field is zero within the solid part of the conductor
• the electric field at the surface of the conductor is perpendicular to the surface
• charge accumulates, and the field is strongest, on pointy parts of the conductor

Let’s see if we can explain these things. Consider a negatively-charged conductor; in other words, a conductor with an excess of electrons. The excess electrons repel each other, so they want to get as far away from each other as possible. To do this they move to the surface of the conductor. They also distribute themselves so the electric field inside the conductor is zero. If the field wasn’t zero, any electrons that are free to move would. There are plenty of free electrons inside the conductor (they’re the ones that are canceling out the positive charge from all the protons) and they don’t move, so the field must be zero.
A similar argument explains why the field at the surface of the conductor is perpendicular to the surface. If it wasn’t, there would be a component of the field along the surface. A charge experiencing that field would move along the surface in response to that field, which is inconsistent with the conductor being in equilibrium.
Why does charge pile up at the pointy ends of a conductor? Consider two conductors, one in the shape of a circle and one in the shape of a line. Charges are distributed uniformly along both conductors. With the circular shape, each charge has no net force on it, because there is the same amount of charge on either side of it and it is uniformly distributed. The circular conductor is in equilibrium, as far as its charge distribution is concerned.

With the line, on the other hand, a uniform distribution does not correspond to equilbrium. If you look at the second charge from the left on the line, for example, there is just one charge to its left and several on the right. This charge would experience a force to the left, pushing it down towards the end. For charge distributed along a line, the equilibrium distribution would look more like this:

The charge accumulates at the pointy ends because that balances the forces on each charge

More on Electric field

Electric field

To help visualize how a charge, or a collection of charges, influences the region around it, the concept of an electric field is used. The electric field E is analogous to g, which we called the acceleration due to gravity but which is really the gravitational field. Everything we learned about gravity, and how masses respond to gravitational forces, can help us understand how electric charges respond to electric forces.
The electric field a distance r away from a point charge Q is given by:
Electric field from a point charge : E = k Q / r²
The electric field from a positive charge points away from the charge; the electric field from a negative charge points toward the charge. Like the electric force, the electric field E is a vector. If the electric field at a particular point is known, the force a charge q experiences when it is placed at that point is given by :
F = qE
If q is positive, the force is in the same direction as the field; if q is negative, the force is in the opposite direction as the field

What does an electric field look like?

An electric field can be visualized on paper by drawing lines of force, which give an indication of both the size and the strength of the field. Lines of force are also called field lines. Field lines start on positive charges and end on negative charges, and the direction of the field line at a point tells you what direction the force experienced by a charge will be if the charge is placed at that point. If the charge is positive, it will experience a force in the same direction as the field; if it is negative the force will be opposite to the field.
The fields from isolated, individual charges look like this:

When there is more than one charge in a region, the electric field lines will not be straight lines; they will curve in response to the different charges. In every case, though, the field is highest where the field lines are close together, and decreases as the lines get further apart.

An example

Two charges are placed on the x axis. The first, with a charge of +Q, is at the origin. The second, with a charge of -2Q, is at x = 1.00 m. Where on the x axis is the electric field equal to zero?

This question involves an important concept that we haven’t discussed yet: the field from a collection of charges is simply the vector sum of the fields from the individual charges. To find the places where the field is zero, simply add the field from the first charge to that of the second charge and see where they cancel each other out.
In any problem like this it’s helpful to come up with a rough estimate of where the point, or points, where the field is zero is/are. There is no such point between the two charges, because between them the field from the +Q charge points to the right and so does the field from the -2Q charge. To the right of the -2Q charge, the field from the +Q charge points right and the one from the -2Q charge points left. The field from the -2Q charge is always larger, though, because the charge is bigger and closer, so the fields can’t cancel. To the left of the +Q charge, though, the fields can cancel. Let’s say the point where they cancel is a distance x to the left of the +Q charge.

Cross-multiplying and expanding the bracket gives:

Solving for x using the quadratic equation gives: x = 2.41 m or x = -0.414 m
The answer to go with is x = 2.41 m. This corresponds to 2.41 m to the left of the +Q charge. The other point is between the charges. It corresponds to the point where the fields from the two charges have the same magnitude, but they both point in the same direction there so they don’t cancel out.

Electric Field Line

	Electric field lines can be drawn using field lines. They are also called force lines.
	(Positive charge electric field)	The field lines are originated from the positive charge.	(Negative charge electric field)	The field lines end up at the negative charge.

A positive charge exerts out and a negative charge exerts in equally to all directions; it is symetric. Field lines are drawn to show the direction and strength of field. The closer the lines are, the stronger the force acts on an object. If the lines are further each other, the strength of force acting on a object is weaker.

Electric Field

The electric field (E) is derived in the same way from the equation (see right)
where:

• (constant),
• Q = electric force of one object (C),
• q = electric force of the other object (C), and
• d = distance between the two objects (m).

However, electric field E is a little bit different from gravitational field g. Gravitational force depends on mass, whereas electric force does not depend on mass. Instead, electric force depends on charges on both objects.
By rearranging the formula, we get:

• Electric field (E) for Q:
• Electric field for q:
•  

Let’s divide the electric force (F) by charge q:
Therefore, the electric field tells us the force per unit charge.

Electric Charge

AIEEE 2011 SOLUTIONS-MATHS

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Implications of Gauss’ Law

Gauss’ Law is a powerful method of calculating electric fields. If you have a solid conducting sphere (e.g., a metal ball) that has a net charge Q on it, you know all the excess charge lies on the outside of the sphere. Gauss’ law tells us that the electric field inside the sphere is zero, and the electric field outside the sphere is the same as the field from a point charge with a net charge of Q. That’s a pretty neat result.

The result for the sphere applies whether it’s solid or hollow. Let’s look at the hollow sphere, and make it more interesting by adding a point charge at the center.

What does the electric field look like around this charge inside the hollow sphere? How is the negative charge distributed on the hollow sphere? To find the answers, keep these things in mind:

• The electric field must be zero inside the solid part of the sphere
• Outside the solid part of the sphere, you can find the net electric field by adding, as vectors, the electric field from the point charge alone and from the sphere alone

We know that the electric field from the point charge is given by kq / r². Because the charge is positive, the field points away from the charge.
If we took the point charge out of the sphere, the field from the negative charge on the sphere would be zero inside the sphere, and given by kQ / r² outside the sphere.
The net electric field with the point charge and the charged sphere, then, is the sum of the fields from the point charge alone and from the sphere alone (except inside the solid part of the sphere, where the field must be zero). This is shown in the picture:

How is the charge distributed on the sphere? The electrons must distribute themselves so the field is zero in the solid part. This means there must be -5 microcoulombs of charge on the inner surface, to stop all the field lines from the +5 microcoulomb point charge. There must then be +2 microcoulombs of charge on the outer surface of the sphere, to give a net charge of -5+2 = -3 microcoulombs.

AIEEE 2012 Mathematics – II

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Cartesian system of rectangular co-ordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines. Circles, conic sections Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.

http://rcm.amazon.com/e/cm?t=widgetsamazon-20&o=1&p=8&l=bpl&asins=0691083045&fc1=000000&IS2=1&lt1=_blank&m=amazon&lc1=0000FF&bc1=000000&bg1=FFFFFF&f=ifrUNIT 12:  Three Dimensional Geometry: 
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.

UNIT 13:  Vector Algebra: 
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.

UNIT 14:  STATISTICS AND PROBABILITY: 
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data. Calculation ofstandard deviation, variance and mean deviation for grouped and http://rcm.amazon.com/e/cm?t=widgetsamazon-20&o=1&p=8&l=bpl&asins=0071544259&fc1=000000&IS2=1&lt1=_blank&m=amazon&lc1=0000FF&bc1=000000&bg1=FFFFFF&f=ifrungrouped data.
Probability: Probability of an event, addition and multiplication theorems of probability, Baye’s theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution.

UNIT 15:  Trigonometry: 
Trigonometrical identities and equations. Trigonometrical functions. Inverse trigonometrical functions and their properties. Heights and Distances.

UNIT 16:  MATHEMATICAL REASONING:
Statements, logical operations and, or, implies, implied by, if and only if. Understanding of tautology, contradiction, converse and contra positive.

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AIEEE 2012 Mathematics Syllabus – Units 1-10

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Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions. 
 
UNIT 2:  COMPLEX NUMBERS AND QUADRATIC EQUATIONS: 
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.  

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 Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.

UNIT 4:  PERMUTATIONS AND COMBINATIONS:
 Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.

UNIT 5:  MATHEMATICAL INDUCTION:
 Principle of Mathematical Induction and its simple applications.

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Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.

UNIT 7:  SEQUENCES AND SERIES:
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between A.M. and G.M. Sum upto n terms of special series: Sn, Sn2, Sn3. Arithmetico – Geometric progression.

UNIT 8:  LIMIT, CONTINUITY AND DIFFERENTIABILITY:
Real – valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions. Graphs of simple functions. Limits, continuity and differentiability. Differentiation of the sum, difference, product http://rcm.amazon.com/e/cm?t=widgetsamazon-20&o=1&p=8&l=bpl&asins=0546598161&fc1=000000&IS2=1&lt1=_blank&m=amazon&lc1=0000FF&bc1=000000&bg1=FFFFFF&f=ifrand quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order upto two. Rolle’s and Lagrange’s Mean Value Theorems. Applications of derivatives: Rate of change of quantities, monotonic – increasing and decreasingfunctions, Maxima and minima of functions of one variable, tangents and normals.

UNIT 9:  INTEGRAL CALCULUS:
Integral as an anti – derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities.
Evaluation of simple integrals of the type

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UNIT 10:  Differential Equations:
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type:
dy– + p (x) y = q (x) dx

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