Electric Dipole — Revision Notes

⚡ 30-Second Revision

Electric Dipole: — Two equal and opposite charges ($+q, -q$) separated by $2a$.
Dipole Moment: — $vec{p} = q(2vec{a})$ (from $-q$ to $+q$). Unit: $ext{C}cdot\text{m}$.
Electric Field (Axial): — $vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2vec{p}}{r^3}$ (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 ($heta=90^circ$).
Torque in Uniform Field: — $vec{\tau} = vec{p} \times vec{E}$. Magnitude $au = pE sin\theta$. Max at $heta=90^circ$, zero at $heta=0^circ, 180^circ$.
Potential Energy in Uniform Field: — $U = -vec{p} cdot vec{E} = -pE cos\theta$. Min at $heta=0^circ$ (stable), Max at $heta=180^circ$ (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 $rac{1}{4piepsilon_0} \frac{2p}{r^3}$ and points along $vec{p}$. On the equatorial line, it's $rac{1}{4piepsilon_0} \frac{p}{r^3}$ and points opposite to $vec{p}$. The electric potential falls off as $1/r^2$, given by $rac{1}{4piepsilon_0} \frac{p cos\theta}{r^2}$. 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 $vec{E}$ ($heta=0^circ$), and maximum (unstable equilibrium) when anti-aligned ($heta=180^circ$).

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 ($ext{C}cdot\text{m}$). 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 ($r gg a$), $vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2vec{p}}{r^3}$. 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 ($heta=90^circ$) 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 $au = pE sin\theta$. This torque tends to align $vec{p}$ with $vec{E}$. Maximum torque occurs when $heta=90^circ$, and zero torque when $heta=0^circ$ 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 $heta=0^circ$ (aligned) and maximum (unstable equilibrium) when $heta=180^circ$ (anti-aligned).
Work Done: — The work done by an external agent to rotate the dipole from $heta_1$ to $heta_2$ 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 $heta=60^circ$, find torque and potential energy. $au = pE sin\theta = (10^{-9})(2 \times 10^4)sin(60^circ) = (2 \times 10^{-5})(sqrt{3}/2) = sqrt{3} \times 10^{-5},\text{N}cdot\text{m}$. $U = -pE cos\theta = -(10^{-9})(2 \times 10^4)cos(60^circ) = -(2 \times 10^{-5})(1/2) = -10^{-5},\text{J}$.

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 ($ext{C}cdot\text{m}$). It's a vector quantity.

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

* Axial Line (End-on position): $vec{E}_{axial} = \frac{1}{4piepsilon_0} \frac{2vec{p}}{r^3}$. 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 $heta$ is the angle between $vec{p}$ and the position vector $vec{r}$. * Axial Line: $heta=0^circ$ or $180^circ$. $V = pm \frac{1}{4piepsilon_0} \frac{p}{r^2}$. * Equatorial Line: $heta=90^circ$. $V = 0$. 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 $au = pE sin\theta$. * $au_{max}$ when $heta=90^circ$ (dipole perpendicular to field).

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

* $U_{max} = +pE$ when $heta=180^circ$ (unstable equilibrium). * Reference point: $U=0$ at $heta=90^circ$. * 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 $vec{p}$)