divergence(V,X) returns
the divergence
of vector fieldV with respect to the
vector X in Cartesian coordinates. Vectors V and X must
have the same length.

Find the electric charge density for the electric field $$\overrightarrow{E}={x}^{2}\overrightarrow{i}+{y}^{2}\overrightarrow{j}$$.

syms x y ep0
E = [x^2 y^2];
rho = ep0*divergence(E,[x y])

rho =
ep0*(2*x + 2*y)

Visualize the electric field and electric charge density for
-2<x<2 and -2<y<2 with
ep0=1. Create a grid of values of
x and y using
meshgrid. Find the values of electric field and
charge density by substituting grid values using subs.
Simultaneously substitute the grid values xPlot and
yPlot into the charge density rho
by using cells arrays as inputs to subs.

rho = subs(rho,ep0,1);
v = -2:0.1:2;
[xPlot,yPlot] = meshgrid(v);
Ex = subs(E(1),x,xPlot);
Ey = subs(E(2),y,yPlot);
rhoPlot = double(subs(rho,{x,y},{xPlot,yPlot}));

Plot the electric field using quiver. Overlay the
charge density using contour. The contour lines
indicate the values of the charge density.

quiver(xPlot,yPlot,Ex,Ey)
hold on
contour(xPlot,yPlot,rhoPlot,'ShowText','on')
title('Contour Plot of Charge Density Over Electric Field')
xlabel('x')
ylabel('y')

V — Vector field symbolic expression | symbolic function | vector of symbolic expressions | vector of symbolic functions

Vector field to find divergence of, specified as a symbolic
expression or function, or as a vector of symbolic expressions or
functions. V must be the same length as X.

X — Variables with respect to which you find the divergence symbolic variable | vector of symbolic variables

Variables with respect to which you find the divergence, specified
as a symbolic variable or a vector of symbolic variables. X must
be the same length as V.

The divergence of the vector field V = (V_{1},...,V_{n}) with respect to the vector X = (X_{1},...,X_{n}) in Cartesian coordinates is the sum of partial derivatives of
V with respect to
X_{1},...,X_{n}.

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