Recent content by Mike400

Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...

I tried to find it the following way but to no avail: Let maximum value of ##\sigma## be ##S## Now unfortunately, we do not have a maximum value for ##\dfrac{1}{r^2}## because the field point can be as close as we want to the arbitrary surface charge. (The field at a point on the surface is...

##\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}\\ =\lim\limits_{\Delta x \to 0}\dfrac{\displaystyle\int_{V'} \dfrac{(x+\Delta x)-x'}{|\mathbf{r}(x+\Delta x,y,z)-\mathbf{r'}|^3}\ dV' - \int_{V'} \dfrac{x-x'}{|\mathbf{r}(x,y,z)-\mathbf{r'}|^3}\ dV'}{\Delta x}\\...

Are there any other simple methods to prove it without using Gauss law?

Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...

Thanks a lot... Anyway I have to clear some confusions regarding that. I will post my confusions tomorrow... I am so sleepy

Can we prove it using Gauss law?

I know how Gauss law helps us to calculate the discontinuity at a point on the surface of a surface charge. Similarly using Gauss law, is there a way to determine the continuity at other points of electric field due to a surface charge or the continuity at all points of electric field due to a...

Both your equations are correct and that was what I was referring to. Mathematically speaking, it is applicable inside the distribution also. See the below two references: (1) Reflections in Maxwell's treatise. Section 4.2 (2) Electromagnetic theory. (by Alfred) Chapter II Section 2, The para...

Sorry for the confusion. Actually all the integration are with respect to source (primed) coordinates. The equation I mention is reached by taking the negative gradient of potential due to continuous electric dipole distribution.

Let: ##\nabla## denote dell operator with respect to field coordinate (origin) ##\nabla'## denote dell operator with respect to source coordinates The electric field at origin due to an electric dipole distribution in volume ##V## having boundary ##S## is: \begin{align} \int_V...

##\rho## is continuous, bounded and its domain ##V'## is finite. Any more properties required to be added?? I was also not convinced myself with part I and part II in the first place. The convergence theorems seem to be too esoteric to me. Do you know another simpler method to show the...

Can anyone point out the flaws in my work if there are any.

I know that and can be simply deduced from the equation ##\mathbf{B}=\mathbf{H} + \mu_0 \mathbf{M}=\mathbf{H}^{V} + \mathbf{H}^{S} + \mu_0 \mathbf{M}##. But should the tangential component of ##\vec{B}## must be continuous too? The equation shows a discontinuity in the tangential component...