The expression for torque '\(\vec{\tau}\)' experienced by an electric dipole of dipole moment '\(\vec{P}\)' in an external uniform electric field '\(\vec{E}\)' is given by :
Understanding Torque on an Electric Dipole in a Uniform Electric Field
Let's break down the concept of an electric dipole experiencing torque when placed in a uniform external electric field. An electric dipole consists of two equal and opposite charges, +q and -q, separated by a small distance, say 2a. The electric dipole moment, denoted by \(\vec{P}\), is a vector quantity. Its magnitude is \(P = q \times 2a\), and its direction is from the negative charge to the positive charge.
A uniform electric field, \(\vec{E}\), is a region where the electric field strength and direction are the same at all points.
Forces on the Electric Dipole in a Uniform Electric Field
When this electric dipole is placed in a uniform external electric field \(\vec{E}\):
- The charge +q experiences a force \(\vec{F}_{+} = q\vec{E}\) in the direction of the electric field.
- The charge -q experiences a force \(\vec{F}_{-} = -q\vec{E}\) in the direction opposite to the electric field.
The net force on the electric dipole is the vector sum of these two forces:
\(\vec{F}_{net} = \vec{F}_{+} + \vec{F}_{-} = q\vec{E} + (-q\vec{E}) = \vec{0}\)
Since the net force is zero, the dipole does not undergo any translatory motion (motion in a straight line or curve). However, the two forces, \(\vec{F}_{+}\) and \(\vec{F}_{-}\), are equal in magnitude, opposite in direction, and act at different points along the dipole. This forms a couple, which tends to rotate the dipole. This rotational effect is called torque.
Calculating the Torque on the Electric Dipole
The torque (\(\vec{\tau}\)) exerted by a couple is given by the product of the magnitude of one of the forces and the perpendicular distance between the lines of action of the two forces.
Let the electric dipole moment \(\vec{P}\) make an angle \(\theta\) with the direction of the electric field \(\vec{E}\). The distance between the charges is 2a. The magnitude of each force is \(F = qE\).
The perpendicular distance between the lines of action of the forces is \(2a \sin\theta\).
The magnitude of the torque is:
\(\tau = F \times (2a \sin\theta)\)
\(\tau = (qE) \times (2a \sin\theta)\)
\(\tau = (q \times 2a) E \sin\theta\)
Since \(P = q \times 2a\), we have:
\(\tau = PE \sin\theta\)
In vector form, the torque is given by the cross product of the electric dipole moment vector \(\vec{P}\) and the electric field vector \(\vec{E}\).
\(\vec{\tau} = \vec{P} \times \vec{E}\)
The direction of the torque vector \(\vec{\tau}\) is perpendicular to the plane containing \(\vec{P}\) and \(\vec{E}\), and is given by the right-hand rule. This torque tends to align the electric dipole moment \(\vec{P}\) with the direction of the electric field \(\vec{E}\).
Analyzing the Options
Let's compare the derived expression for torque with the given options:
- \(\vec{\tau}=\vec{P}.\vec{E}\): This represents the scalar dot product, which gives a scalar quantity. Torque is a vector quantity. So, this is incorrect.
- \(\vec{\tau}=\frac{\vec{P}}{\vec{E}}\): Division of vectors is not defined in this manner for physical quantities like torque. So, this is incorrect.
- \(\vec{\tau}=\frac{\vec{E}}{\vec{P}}\): Similar to the previous option, this vector division is not physically meaningful in this context. So, this is incorrect.
- \(\vec{\tau}=\vec{P}\times \vec{E}\): This represents the vector cross product. The magnitude is \(PE \sin\theta\) and the direction is perpendicular to \(\vec{P}\) and \(\vec{E}\), which matches our derivation for the torque on an electric dipole.
Therefore, the expression for torque experienced by an electric dipole \(\vec{P}\) in an external uniform electric field \(\vec{E}\) is \(\vec{\tau}=\vec{P}\times \vec{E}\).
Revision Table: Electric Dipole in Electric Field
| Concept | Description | Formula |
|---|---|---|
| Electric Dipole Moment | Vector from -q to +q, magnitude \(q \times 2a\) | \(\vec{P} = q(2\vec{a})\) |
| Forces on Dipole in Uniform Field | Equal and opposite forces \(q\vec{E}\) and \(-q\vec{E}\) on +q and -q | \(\vec{F}_+ = q\vec{E}\), \(\vec{F}_- = -q\vec{E}\) |
| Net Force on Dipole | Sum of forces | \(\vec{F}_{net} = \vec{0}\) |
| Torque on Dipole | Rotational effect due to the couple | \(\vec{\tau} = \vec{P} \times \vec{E}\) |
| Magnitude of Torque | Magnitude of cross product | \(\tau = PE \sin\theta\) |
Additional Information: Potential Energy of an Electric Dipole
Besides experiencing torque, an electric dipole in an external electric field also possesses potential energy. When the dipole rotates from one orientation to another in the field, the electric field does work. The potential energy of the electric dipole is defined as the work done in bringing the dipole from infinity to its current orientation in the electric field.
Alternatively, it can be considered as the negative of the work done by the electric field in rotating the dipole from a reference orientation (usually perpendicular to the field, where potential energy is taken as zero) to the given orientation.
The expression for the potential energy (U) of an electric dipole \(\vec{P}\) in a uniform electric field \(\vec{E}\) is given by the negative of the dot product of \(\vec{P}\) and \(\vec{E}\):
\(U = -\vec{P} \cdot \vec{E}\)
In terms of magnitude and the angle \(\theta\) between \(\vec{P}\) and \(\vec{E}\), this is:
\(U = -PE \cos\theta\)
- The potential energy is minimum (\(-PE\)) when \(\theta = 0^\circ\), meaning the dipole is aligned with the field (stable equilibrium).
- The potential energy is maximum (\(+PE\)) when \(\theta = 180^\circ\), meaning the dipole is anti-aligned with the field (unstable equilibrium).
- The potential energy is zero when \(\theta = 90^\circ\), where \(\vec{P}\) is perpendicular to \(\vec{E}\).
Understanding both torque and potential energy is crucial for analyzing the behavior of electric dipoles in electric fields.
Important Questions from Electric Fields and Gauss' Law
A positive charge +q is placed at the centre of a hollow metallic sphere of inner radius a and outer radius b. the electric field at a distance r from the centre is denoted by E. In this regards, which one of the following statement is correct?
If a free electron moves through a potential difference of 1 kV, then the energy gained by the electron is given by
- Two point charges $q_1 \left( {\sqrt {10} {\rm{\mu C}}} \right)$ and $q_2(-18\sqrt{2} {\rm{\mu C}})$ are placed on the x-axis at $x = 0$ m and $x = 4$ m respectively. The electric field (in V/m) at a point $(1, 3)$ m is,
$\left[ {{\rm{Take\;}}\frac{1}{{4{\rm{\pi }}{\epsilon_0}}} = 9 \times {{10}^9}{\rm{N}}{{\rm{m}}^2}{{\rm{C}}^{ - 2}}} \right]$ - Let a total charge $2Q$ be distributed in a sphere of radius $R$, with the charge density given by $\rho(r) = Cr^2$, where $r$ is the distance from the centre. Two charges $A$ and $B$, of $-Q$ each, are placed on diametrically opposite points, at equal distance, '$a$' from the centre. If $A$ and $B$ do not experience any force, then:
The surface charge density of a thin spherical shell placed in an air medium is 88.54 c/m2 The intensity of the electric field measured 12 mm outside the shell from the centre of the shell is 5.625 × 101 2 N/C. The thin spherical shell has a radius of:
