Electrostatics Concept Page - 5

Definition

Density of electric field lines

Magnitude of electric field strength is higher where density of electric field lines in space is more. The attached figure shows the electric field lines due to a positive point charge. Its field lines are directed radially outwards from the point charge and the dotted lines show the locus of points with same magnitude of electric field.
Since, E∝1r2$E\propto {\displaystyle \frac{1}{r^2}}$ for a point charge, it is larger near the charge. This behaviour is consistent in the electric field lines of point charge.

Definition

Physical significance of electric field

Electric field is an elegant way of characterising the electrical environment of a system of charges. Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field.
Suppose we consider the force between two distant charges q1,q2$q_1,q_2$ in accelerated motion.There will be some time delay between the effect(force on q2$q_2$) and the cause (motion of q1$q_1$).
The field picture is, the accelerated motion of charge q1$q_1$ produces electromagnetic waves, which then propagate with the speed 'c', reach q2$q_2$ and cause a force on q2$q_2$. The notion of field elegantly accounts for the time delay.

Definition

Electric field intensity as a function of distance for a charged spherical shell

E=0     (r<a)$E=0(r<a)$
E=kQr2     (r>a)$E={\displaystyle \frac{kQ}{r^2}}(r>a)$
where a is the radius of the shell.

Definition

Non uniform charge distribution

If charge is non uniformly spread over the surface then it is non uniform distribution of charges.

Definition

Uniform Charge distribution

 When the charge is not accumulated in some part but is spread uniformly then it is called uniform charge distribution

Definition

Explain continuous distribution of charges

In a continuous charge distribution, all the charges are closely bound together i.e. having very less space between them. But this closely bound system doesnt means that the electric charge is uninterrupted. It clears that the distribution of separate charges is continuous, having a minor space between them.

Example

Electric field due to a continuous charge distribution

Example:
A rod of length l$l$ with uniform charge per unit length λ$\lambda$ is placed at a distance d$d$ from origin along the x-axis. A similar rod is placed at the same distance along Y-axis. Determine the magnitude of net electric field intensity at the origin.
Solution:Let's first find Electric field due to the rod on x-axis, say E1$E_1$
Consider a small length element dx$dx$, distance x$x$ away from origin.
Field  due to this dE1=kλdxx2$d{E}_1={\displaystyle \frac{k\lambda dx}{x^2}}$
Integrating from x=d$x=d$ to x=d+l$x=d+l$
GivesE1=λl4πϵ0d(d+l)$E_1={\displaystyle \frac{\lambda l}{4\pi {ϵ}_{0}d(d+l)}}$ along -x direction
Now, E2$E_2$is along the -y direction and same in magnitude.
Thus,
|E|=2√|E1|=2√λl4πϵ0d(d+l)$|E|=\sqrt{2}|E_1|={\displaystyle \frac{\sqrt{2}\lambda l}{4\pi {ϵ}_{0}d(d+l)}}$

Definition

Charge density

In electromagnetism, charge density is a measure of electric charge per unit volume of space, in one, two or three dimensions.
A surface charge density σ$\sigma$ at an area element is given by
σ=ΔQΔA$\sigma ={\displaystyle \frac{\mathrm{\Delta }Q}{\mathrm{\Delta }A}}$
where ΔQ$\mathrm{\Delta }Q$ is the charge and ΔA$\mathrm{\Delta }A$ is the area element.
The linear charge density λ$\lambda$ of a wire is defined by
λ=ΔQΔl$\lambda ={\displaystyle \frac{\mathrm{\Delta }Q}{\mathrm{\Delta }l}}$
where λ$\lambda$ is the linear charge density, Δl$\mathrm{\Delta }l$ is a small line element of wire.
The volume charge density is defined by 
ρ=ΔQΔV$\rho ={\displaystyle \frac{\mathrm{\Delta }Q}{\mathrm{\Delta }V}}$
where ΔV$\mathrm{\Delta }V$ is an elemental volume and ρ$\rho$ is volume charge density.

Definition

Electric field

The electric field due to a charge Q$Q$ at a point in space may be defined as the force that a unit positive charge would experience if placed at that point. The charge Q$Q$, which is producing the electric field, is called a source charge and the charge q$q$, which tests the effect of a source charge, is called a test charge. 
E=limq→0(Fq)=kQr2$E=\underset{q\to 0}{lim}(\frac{F}{q})={\displaystyle \frac{kQ}{r^2}}$ where r is the distance from the source charge where electric field is calculated.

Definition

Force on a charged body in an electric field

Force on charge q$q$ in electric field E⃗ $\overrightarrow{E}$ is given by:
F⃗ =qE⃗ $\overrightarrow{F}=q\overrightarrow{E}$
Example: 
An electron and proton are placed in an electric field. The forces acting on them are F1$F_1$ and F2$F_2$ and their accelerations are a1$a_1$ and a2$a_2$ respectively, thenF1=−F2=−eE$F_1=-F_2=-eE$
F1+F2=0$F_1+F_2=0$

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