Electrostatics Concept Page - 13

Example

Electric field of an electric dipole for equatorial points

Two charges each of 10 C are placed 5.0 mm apart.Determine the electric field at a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole,E=p4πϵ0r3(r/a>>1)$E={\displaystyle \frac{p}{4\pi {ϵ}_0{r}^3}}(r/a>>1)$
=5×10−84π×(8.854×10−12C2N−1m−2)×1(15)3×10−6m3$={\displaystyle \frac{5\times {10}^{-8}}{4\pi \times (8.854\times {10}^{-12}{C}^2{N}^{-1}{m}^{-2})}}\times {\displaystyle \frac{1}{(15{)}^3\times {10}^{-6}{m}^3}}$
=1.33×105NC−1$=1.33\times {10}^{5}N{C}^{-1}$
The direction of electric field in this case is opposite to the direction of the dipole moment vector.

Formula

Electric field due to a dipole

E=14πϵ0pr33cos2θ+1−−−−−−−−√$E={\displaystyle \frac{1}{4\pi {ϵ}_0}}{\displaystyle \frac{p}{r^3}}\sqrt{3co{s}^2\theta +1}$
where θ$\theta$ is the  angle between the distance vector and dipole.

Definition

Potential due to a dipole at a general point

Potential due to a dipole is given by:
V≈pcosθ4πεor2$V\approx {\displaystyle \frac{p\cos \theta }{4\pi {\epsilon }_o{r}^2}}$
where 
p:$p:$ electric dipole moment
θ:$\theta :$ angle made by the line joining center of dipole and the point with the dipole moment vector
r:$r:$ distance between the center of dipole and the point
Note:
Approximation is made that the length of dipole is negligible as compared to the distance of the point from the dipole.

Definition

Potential due to an electric dipole

V=14πϵ0pcosθr2$V={\displaystyle \frac{1}{4\pi {ϵ}_0}}{\displaystyle \frac{pcos\theta }{r^2}}$
where p is the dipole moment, r is the distance at which the potential V of the dipole is calculated and θ$\theta$ is the angle between the distance vector and the dipole.

Definition

Potential energy of a dipole in a uniform electric field

U(θ)=−p⃗ .E⃗ $U(\theta )=-\overrightarrow{p}.\overrightarrow{E}$
where p⃗ $\overrightarrow{p}$ is the dipole moment, E⃗ $\overrightarrow{E}$ is the electric field and θ$\theta$ is the angle between dipole moment and electric field.

Definition

Self potential energy of a dipole

Self potential energy of a dipole is the total binding energy of a dipole. Hence, it is given by:
U=−Q24πεod$U={\displaystyle \frac{-Q^2}{4\pi {\epsilon }_{o}d}}$
∴U=−p24πεod3$\therefore U={\displaystyle \frac{-p^2}{4\pi {\epsilon }_o{d}^3}}$
where:
Q:$Q:$ charge
d:$d:$ length of dipole
p:$p:$ dipole moment

Example

Work done in rotating an electric dipole in an electric field

An electric dipole is along a uniform electric field. If it is deflected by 600${60}^0$, work done by the agent is 2×10−19 J$2\times {10}^{-19}J$. Then the work done by an agent if it is deflected by 300${30}^0$ further is:
If P$P$ be the dipole moment and E$E$ be the electric field then the potential energy of a dipole is given by:
W=P.E.cosθ$W=P.E.\cos \theta$
Now, given: P.E.cos60o=2×10−19 J$P.E.\cos {60}^o=2\times {10}^{-19}J$
Then, P.E=4×10−19 J$P.E=4\times {10}^{-19}J$
Now, the work done to move the dipole further by 30o${30}^o$,
W=P.E.cos90o−P.E.cos60o$W=P.E.\cos {90}^o-P.E.\cos {60}^o$=0−(4×10−19×12) J$=0-(4\times {10}^{-19}\times \frac{1}{2})J$
or, |W|=2×10−19 J$|W|=2\times {10}^{-19}J$

Definition

Torque on a dipole in a uniform electric field

An electric dipole consists of two opposite charges of magnitude 1μC$1\mu C$ separated by a distance of 2cm. The dipole is placed in an electric field 10−5Vm−1${10}^{-5}V{m}^{-1}$. The maximum torque that the field exerts on the dipole is:The torque experienced by a dipole in an electric field E is p⃗ ×E⃗ $\overrightarrow{p}\times \overrightarrow{E}$
Maximum torque =pE=qaE=10−6×0.02×10−5=2×10−13 Nm$=pE=qaE={10}^{-6}\times 0.02\times {10}^{-5}=2\times {10}^{-13}Nm$

Formula

Force per unit area of a charged conductor

p=σEext=σ22ϵ0=ϵ02E2$p=\sigma E_{ext}={\displaystyle \frac{{\sigma }^2}{2{ϵ}_0}}={\displaystyle \frac{{ϵ}_0}{2}}{E}^2$
where p is the force per unit area, σ$\sigma$ is the surface charge density

Definition

Electrostatic field is zero inside a conductor

Consider a conductor, neutral or charged. There may also be an external electrostatic field.
In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor.
A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside. Electrostatic field is zero inside a conductor.

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