I) Gradient
II) Divergence
III) Curl and its physical interpretation
Answer: I)Gradient: For a realvalued function f(x,y,z)f(x,y,z) on R3, the gradient ∇f(x,y,z)∇f(x,y,z) is a vectorvalued function of R3, that is, its value at a point (x,y,z)(x,y,z) is the vector∇f(x,y,z)=(∂f/∂x,∂f/∂y,∂f/∂z)=∂f/∂x i+∂f/∂y j+∂f/∂zk ∇f(x,y,z)=(∂f/∂x,∂f/∂y,∂f/∂z)=∂f/∂x i+∂f/∂y j+∂/f∂z kIn R3, where each of the partial derivatives is evaluated at the point (x,y,z)(x,y,z). So in this way, you can think of the symbol ∇ as being “applied” to a realvalued function ff to produce a vector ∇f.It turns out that the divergence and curl can also be expressed in terms of the symbol ∇. This is done by thinking of ∇ as a vector in R3, namely ∇=∂/∂x i+∂/∂y j+∂/∂z k.II) Divergenceit is often convenient to write the divergence div f as ∇⋅f, since for a vector field f(x,y,z)=f1(x,y,z)i+f2(x,y,z)j+f3(x,y,z)kf(x,y,z)=f1(x,y,z)i+f2(x,y,z)j+f3(x,y,z)k, The dot product of f with ∇ (thought of as a vector) makes sense:∇⋅f=(∂/∂xi+∂/∂yj+∂/∂zk)⋅(f1(x,y,z)i+f2(x,y,z)j+f3(x,y,z)k)=(∂/∂x)(f1)+(∂/∂y)(f2)+(∂/∂z)(f3)=∂f1/∂x+∂f2/∂y+∂f3/∂z==div fIII) Curl and its physical interpretation: The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,Physical Interpretation:The curl of a vector field measures the tendency for the vector field to swirl around. Imagine that the vector field represents the velocity vectors of water in a lake. If the vector field swirls around, then when we stick a paddle wheel into the water, it will tend to spin.Question: What is the various types of charge distributions?Answer:Type of charge distribution  Denoted by  Unit 
Line Charge  λ (Line charge density)  C/m 
Surface Charge  σ (surface charge density)  C/m2 
Volume Charge  ρ (volume charge density)  C/m3 
Consider a point charge q called SOURCE CHARGE placed at a point ‘O’ in space. To find its intensity at a point ‘p’ at a distance ‘r’ from the point charge we place a test charge 'q'.  
 The force experienced by the test charge q’ will be:  
 F = Eq’(1)  
 According to coulomb's law the electrostatic force between them is given by:  
 
 Putting the value of 'F' we get : www.citycollegiate.com  

 

 
 This shows that the electric intensity due to a point charge is directly proportional to the magnitude of charge q and inversely proportional to the square of distance  
“Electric Field Due to an Infinite Line Charge using Gauss’ Law,” where we found E=ρ^ρl2πϵsρ This is a consequence of Gauss’ Law ∮SD⋅ds=Qencl V=−∫CE⋅dl ) Wrapping up: C≜Q+/V=ρll(ρl/2πϵs)ln(D/d) Note that factors of ρlρl in the numerator and denominator cancel out, leaving: C=2πϵsl / (ln(b/a)) 
Laplace’s equation is a linear, homogeneous, partial differential equation. It has the form:∇2u=0or if u∈Rn then:∂2u/∂x12+……....+∂2u/∂xn2=0Poisson’s equation is simply the inhomogeneous version of Laplace’s equation. That means it is of the form:∂2u/∂x12+……....+∂2u/∂xn2=f(x1,...,xn)for u∈Rn f∈C(Rn)As a final note I am not sure about whether there is an strict, universal definition on the source function ff. I have gone with the one I have seen used most often but I am interested if anyone knows whether there are ever more relaxed requirements for f. Question: Find H at the centre of the square loop of side L in xy plane at the origin as centre, carrying current I.Answer: H=4[2μoI(sin45o+sin45o)/ 4πL] =2root(2) μoI/ πaQuestion: Calculate the internal and external inductance per unit length of a transmission line consisting of two long parallel conducting wires of radius a that carry current in opposite directions the axes of the wires are separated by a distance d, which is much larger than a.Answer:
The intrinsic impedance is a property of a medium  an area of space. For a vacuum (outer space) or for wave propagation through the air around earth (often called 'free space'), the intrinsic impedance (often written as or Z) is given by: = This parameter is the ratio of the magnitude of the Efield to the magnitude of the Hfield for a plane wave in a lossless medium (zero conductivity): This relation can be derived directly from Maxwell's Equations. For a general medium with permittivity and permeability given by , the intrinsic impedance is given by: For a medium with a conductivity associated with it, the intrinsic impedance is given by: When the conductivity is nonzero, the above intrinsic impedance is a complex number, indicating that the electric and magnetic fields are not inphase. The intrinsic impedance of freespace has nothing to do with the electrical impedance of an antenna. Also, there is no reason to have the impedance of an antenna match the intrinsic impedance of free space (no mismatch loss occurs). 