Electric Dipole — Revision Notes

⚡ 30-Second Revision

Electric Dipole: — Two equal and opposite charges ($+q, -q$) separated by $2a$.
Dipole Moment: — $\vec{p} = q(2\vec{a})$ (from $-q$ to $+q$). Unit: $\text{C}\cdot\text{m}$.
Electric Field (Axial): — $\vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2\vec{p}}{r^3}$ (along $\vec{p}$, for $r \gg a$).
Electric Field (Equatorial): — $\vec{E}_{equatorial} = -\frac{1}{4piepsilon_0} \frac{\vec{p}}{r^3}$ (opposite to $\vec{p}$, for $r \gg a$).
Electric Potential: — $V = \frac{1}{4piepsilon_0} \frac{p \cos\theta}{r^2}$ (for $r \gg a$). $V=0$ on equatorial line ($\theta=90^\circ$).
Torque in Uniform Field: — $\vec{\tau} = \vec{p} \times \vec{E}$. Magnitude $\tau = pE \sin\theta$. Max at $\theta=90^\circ$, zero at $\theta=0^circ, 180^\circ$.
Potential Energy in Uniform Field: — $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$. Min at $\theta=0^\circ$ (stable), Max at $\theta=180^\circ$ (unstable).
Work Done: — $W = U_{final} - U_{initial}$.
Net Charge: — Zero.
Force in Uniform Field: — Zero net force.

2-Minute Revision

An electric dipole is a system of two equal and opposite charges, $+q$ and $-q$, separated by a small distance $2a$. Its defining characteristic is the electric dipole moment, $\vec{p}$, a vector from $-q$ to $+q$ with magnitude $q(2a)$. Remember, the net charge of a dipole is zero.

The electric field due to a dipole falls off as $1/r^3$, unlike a single charge's $1/r^2$. On the axial line, the field is $\frac{1}{4piepsilon_0} \frac{2p}{r^3}$ and points along $\vec{p}$. On the equatorial line, it's $\frac{1}{4piepsilon_0} \frac{p}{r^3}$ and points opposite to $\vec{p}$. The electric potential falls off as $1/r^2$, given by $\frac{1}{4piepsilon_0} \frac{p \cos\theta}{r^2}$. Crucially, the potential is zero everywhere on the equatorial line.

When placed in a uniform electric field $\vec{E}$, a dipole experiences a torque $\vec{\tau} = \vec{p} \times \vec{E}$, which tries to align $\vec{p}$ with $\vec{E}$. The magnitude is $pE \sin\theta$, maximum at $90^\circ$.

The net force on the dipole in a uniform field is zero. Its potential energy is $U = -pE \cos\theta$, minimum (stable equilibrium) when $\vec{p}$ is aligned with $\vec{E}$ ($\theta=0^\circ$), and maximum (unstable equilibrium) when anti-aligned ($\theta=180^\circ$).

Work done in rotation is the change in potential energy.

5-Minute Revision

Let's consolidate the key aspects of electric dipoles for NEET. An electric dipole is fundamentally a pair of equal and opposite charges, $+q$ and $-q$, separated by a fixed distance $2a$. The most important quantity is the electric dipole moment $\vec{p}$, a vector of magnitude $q(2a)$ directed from $-q$ to $+q$. Its SI unit is Coulomb-meter ($\text{C}\cdot\text{m}$). Remember, the net charge of a dipole is always zero.

Electric Field and Potential: Unlike a single charge, a dipole's field and potential decrease faster with distance. The electric field $\vec{E}$ falls off as $1/r^3$, and the electric potential $V$ as $1/r^2$.

Axial Line: — At a distance $r$ from the center along the axis ($r \gg a$), $\vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2\vec{p}}{r^3}$. The field is in the same direction as $\vec{p}$.
Equatorial Line: — At a distance $r$ from the center along the perpendicular bisector ($r \gg a$), $\vec{E}_{equatorial} = -\frac{1}{4piepsilon_0} \frac{\vec{p}}{r^3}$. The field is opposite to $\vec{p}$.
General Point: — The potential at a point $(r, \theta)$ is $V = \frac{1}{4piepsilon_0} \frac{p \cos\theta}{r^2}$. A critical point: potential on the equatorial line ($\theta=90^\circ$) is always zero.

Dipole in an External Electric Field:

Uniform Field: — If placed in a uniform electric field $\vec{E}$, the dipole experiences zero net force because the forces on $+q$ and $-q$ are equal and opposite. However, it experiences a torque $\vec{\tau} = \vec{p} \times \vec{E}$, with magnitude $\tau = pE \sin\theta$. This torque tends to align $\vec{p}$ with $\vec{E}$. Maximum torque occurs when $\theta=90^\circ$, and zero torque when $\theta=0^\circ$ or $180^\circ$.
Potential Energy: — The potential energy of the dipole in a uniform field is $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$. It's minimum (stable equilibrium) when $\theta=0^\circ$ (aligned) and maximum (unstable equilibrium) when $\theta=180^\circ$ (anti-aligned).
Work Done: — The work done by an external agent to rotate the dipole from $\theta_1$ to $\theta_2$ is $W = U_2 - U_1 = -pE(\cos\theta_2 - \cos\theta_1)$.
Non-Uniform Field: — In a non-uniform field, the dipole experiences both a net force and a torque, as the forces on $+q$ and $-q$ are no longer equal in magnitude.

Example: A dipole with $p = 10^{-9},\text{C}\cdot\text{m}$ is in a $2 \times 10^4\,\text{N/C}$ field. If $\theta=60^\circ$, find torque and potential energy. $\tau = pE \sin\theta = (10^{-9})(2 \times 10^4)\sin(60^\circ) = (2 \times 10^{-5})(\sqrt{3}/2) = \sqrt{3} \times 10^{-5},\text{N}\cdot\text{m}$. $U = -pE \cos\theta = -(10^{-9})(2 \times 10^4)\cos(60^\circ) = -(2 \times 10^{-5})(1/2) = -10^{-5},\text{J}$.

Prelims Revision Notes

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Definition: — An electric dipole consists of two point charges, $+q$ and $-q$, of equal magnitude but opposite sign, separated by a small distance $2a$. The net charge of a dipole is always zero.
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**Electric Dipole Moment ($\vec{p}$):**

* Magnitude: $p = q(2a)$. * Direction: From $-q$ to $+q$. * Unit: Coulomb-meter ($\text{C}\cdot\text{m}$). It's a vector quantity.

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**Electric Field due to a Dipole (for $r \gg a$):**

* Axial Line (End-on position): $\vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2\vec{p}}{r^3}$. Direction is along $\vec{p}$. * Equatorial Line (Broadside-on position): $\vec{E}_{equatorial} = -\frac{1}{4piepsilon_0} \frac{\vec{p}}{r^3}$. Direction is opposite to $\vec{p}$. * General Point: The field falls off as $1/r^3$. Note that $E_{axial}$ is twice $E_{equatorial}$ in magnitude at the same distance $r$.

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**Electric Potential due to a Dipole (for $r \gg a$):**

* General Point: $V = \frac{1}{4piepsilon_0} \frac{p \cos\theta}{r^2}$, where $\theta$ is the angle between $\vec{p}$ and the position vector $\vec{r}$. * Axial Line: $\theta=0^\circ$ or $180^\circ$. $V = \pm \frac{1}{4piepsilon_0} \frac{p}{r^2}$. * Equatorial Line: $\theta=90^\circ$. $V = 0$. The potential falls off as $1/r^2$.

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**Dipole in a Uniform Electric Field ($\vec{E}$):**

* Net Force: Zero ($\vec{F}_{net} = qvec{E} + (-q)\vec{E} = 0$). * Torque: $\vec{\tau} = \vec{p} \times \vec{E}$. Magnitude $\tau = pE \sin\theta$. * $\tau_{max}$ when $\theta=90^\circ$ (dipole perpendicular to field).

* $\tau_{min}=0$ when $\theta=0^\circ$ or $180^\circ$ (dipole aligned/anti-aligned). * Potential Energy: $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$. * $U_{min} = -pE$ when $\theta=0^\circ$ (stable equilibrium).

* $U_{max} = +pE$ when $\theta=180^\circ$ (unstable equilibrium). * Reference point: $U=0$ at $\theta=90^\circ$. * Work Done by External Agent: $W = U_{final} - U_{initial} = -pE(\cos\theta_f - \cos\theta_i)$.

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Dipole in a Non-Uniform Electric Field: — Experiences both a net force and a torque.
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Key Approximations: — All formulas for field and potential are valid for $r \gg a$ (distance from center much larger than half the separation distance).

Vyyuha Quick Recall

To remember the direction of the dipole moment and its interaction with the field:

Positive Points Positive (from negative to positive) Torque Tries to Turn (align $\vec{p}$ with $\vec{E}$) Axial Along, Equatorial Exactly Opposite (field direction relative to $\vec{p}$)