ELECTROSTATS continu.....

5. Principle Of Superposition

• Coulomb's law gives the electric force acting between two electric charges.
  
  
• Principle of superposition gives the method to find force on a charge when system consists of large number of charges.
  
  
• According to this principle when a number of charges are interacting the total force on a given charge is vector sum of forces exerted on it by all other charges.
  
  
• This principle makes use of the fact that the forces with which two charges attract or repel one another are not affected by the presence of other charges.
  
  
• If a system of charges has n number of charges say q₁, q₂, ...................., q_n, then total force on charge q₁ according to principle of superposition is
  F = F₁₂ + F₁₃ + .................................. F_1n
  Where F₁₂ is force on q₁ due to q₂ and F₁₃ is force on q₁ due to q₃ and so on.
  
  
  
  
  
• F₁₂, F₁₃, .................. F_1n can be calculated from Coulomb's law i. e.
  
  F12=kq1q2rˆ124πϵ0r212
  
  to,
  
  F1n=kq1q2rˆ1n4πϵ0r21n
  
  
• The total force F₁ on the charge q₁ due to all other charges is the vector sum of the forces F₁₂, F₁₃, ................................. F_1n.
  F₁ = F₁₂ + F₁₃ + ..................................
  
  
• The vector sum is obtained by parellogram law of addition of vector.
  
  
• Similarly force on any other charge due to remaining charges say on q₂, q₃ etc. can be found by adopting this method.

6. Electric Field

• Electrical interaction between charged particles can be reformulated using the concept of electric field.
  
  
• To understand the concept consider the mutual repulsion of two positive charged bodies as shown in fig (a)
  
  
  
  
• Now if remove the body B and label its position as point P as shown in fig (b), the charged body A is said to produce an electric field at that point (and at all other points in its vicinity)
  
  
• When a body B is placed at point P and experiences force F, we explain it by a point of view that force is exerted on B by the field not by body A itself.
  
  
• The body A sets up an electric field and the force on body B is exerted by the field due to A.
  
  
• An electric field is said to exists at a point if a force of electric origin is exerted on a stationary charged (test charge) placed at that point.
  
  
• If F is the force acting on test charge q placed at a point in an electric field then electric field at that point is
  E = F/q
  or F = qE
  
  
• Electric field is a vector quantity and since F = qE the direction of E is the direction of F.
  
  
• Unit of electric field is (N.C^-1)
  

Q. Find the dimensions of electric field
Ans. [MLT^-3A^-1]

7. Calculation of Electric Field

• In previous section we studied a method of measuring electric field in which we place a small test charge at the point, measure a force on it and take the ratio of force to the test charge.
  
  
• Electric field at any point can be calculated using Coulomb's law if both magnitude and positions of all charges contributing to the field are known.
  
  
• To find the magnitude of electric field at a point P, at a distance r from the point charge q, we imagine a test charge q'to be placed at P. Now we find force on charge q' due to q through Coulomb's law.
  
  F=kqq′4πϵ0r2
  
  electric field at P is
  E=kqq′4πϵ0r2
  
  The direction of the field is away from the charge q if it is positive
  
  
  
  
• Electric field for either a positive or negative charge in terms of unit vector r directed along line from charge q to point P is
  F=kqq′rˆ4πϵ0r2
  
  r = distance from charge q to point P.
  
  
• When q is negative , direction of E is towards q, opposite to r.
  
  Electric Field Due To Multiple Charges
  
  
• Consider the number of point charges q₁, q₂,........... which are at distance r_1P, r_2P,................... from point P as shown in fig
  
  
  
  
• The resultant electric field is the vector sum of individual electric fields as
  E = E_1P + E_2P + .....................
  
  
  This is also a direct result of principle of superposition discussed earlier in case of electric force on a single charge due to system of multiple charges.
  
  
• E is a vector quantity that varies from one point in space to another point and is determined from the position of square charges.

8. Electric Field Lines

• For a single positive point charge q, electric field is
  
  F=kqq′rˆ4πϵ0r2
  
  now to get feel of this field one can sketch a few representative vectors as shown in fig below
  
  
  
  
• Since electric field varies as inverse of square of the distance that points from the charge the vector gets shorter as you go away from the origin and they always points radially outwards.
  
  
• Connecting up these vectors to form a line is a nice way to represent this field .
  
  
• The magnitude of the field is indicated by the density of the field lines.
  
  
• Magnitude is strong near the center where the field lines are close togather, and weak farther out, where they are relatively apart.
  
  
• So, electric field line is an imaginary line drawn in such a way that it's direction at any point is same as the direction of field at that point.
  
  
• An electric field line is, in general a curve drawn in such a way that the tangent to it ateach point is the direction of net field at that point.
  
  
• Field lines of a single position charge points radially outwards while that of a negative charge are radially inwards as shown below in the figure.
  
  
  
  
• Field lines around the system of two positive charges gives a different picture and describe the mutual repulsion between them.
  
  
  
  
• Field lines around a system of a positive and negative charge clearly shows the mutual attraction between them as shown below in the figure.
  
  
  
  
• Some important general properties of field lines are
  1.Field lines start from positive charge and end on a negative charge.
  2.Field lines never cross each other if they do so then at the point of intersection there will be two direction of electric field.
  3.Electric field lines do not pass through a conductor , this shows that electric field inside a conductor is always zero.
  4.Electric field lines are continuous curves in a charge free region.
  

9. Electric Flux

• Consider a plane surface of area ΔS in a uniform electric field E in the space.
  
  
• Draw a positive normal to the surface and θ be the angle between electric field E and the normal to the plane.
  
  
  
  
• Electric flux of the electric field through the choosen surface is then
  Δφ = E ΔS cosθ
  
  
• Corresponding to area ΔS we can define an area vector ΔS of magnitude ΔS along the positive normal. With this definition one can write electric flux as
  Δφ = E . ΔS
  
  
• direction of area vector is always along normal to the surface being choosen.
  
  
• Thus electric flux is a measure of lines of forces passing through the surface held in the electric field.
  
  Special Cases
  
• If E is perpendicular to the surface i. e., parallel to the area vector then θ = 0 and
  Δφ = E ΔS cos0
  
  
• If θ = π i. e., electric field vector is in the direction opposite to area vector then
  Δφ = - E ΔS
  
  
• If electric field and area vector are perpendicular to each other then θ = π/2 and Δφ = 0
  
  
• Flux is an scaler quantity and it can be added using rules of scaler addition.
  
  
• For calculating total flux through any given surface , divide the surface into small area elements. Calculate the flux at each area element and add them up.
  
  
• Thus total flux φ through a surface S is
  φ ≅ ΣE.ΔS
  
  
• This quantity is mathematically exact only when you take the limit ΔS→0 and the sum in equation 3 is written as integral
  φ = ∫ΣE.dS