Potential Energy in an External Field: Overview, Questions, Preparation

The amount of work required to bring a charge from infinity to a specific position in the field with no change in kinetic energy is defined as potential energy in an external field.

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Formula: U = qV 

where, 

• U is the potential energy 
• V is the potential due to the external field. 
• q is the magnitude of particles. 

Note that the common examples of external fields would be gravitational, electric, and magnetic fields.

 

 

The amount of work done to bring a point of charge from a reference point to the specific point in an electric field, with no acceleration, is the potential energy of a single charge. 

Formula: Potential Energy (U) = q . V 

 Where: 

• The potential energy is U. 

• The electric charge is q. 

• The electric potential at that point is V. 

 Explanation: 

• The electric potential is determined solely by the external charge distribution. 

• A positive charge placed in a positive potential region will have positive potential energy, while a negative charge in the same region will have negative potential. 

• The potential energy is based on the electric potential and the magnitude of the charge. 

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The amount of work done to bring both charges from infinity to their final position in the presence of an external field and considering mutual interaction, is the potential energy of a system of two charges in an external field.

The potential energy of the system is: $U=q_1{V}_1+q_2{V}_2+\frac{1}{4\pi {\epsilon }_0}·\frac{q_1{q}_2}{r_{12}}$
Here,

 

• q1V1 is the potential energy of q1.
• q2V2 is the potential energy of q2.
• $\frac{1}{4\pi {\epsilon }_0}·\frac{q_1{q}_2}{r_{12}}$

Which is the mutual interaction energy here.

The explanation for this goes like this below. 

The first two terms describe the interaction of changes with the external field. They remain independent to each other. The third term comes from the electrostatic forces that charges exert on each other. That's from Coulomb's law. The reference point for zero potential energy becomes convenient and that's why we use it. We need to consider this when both charges are infinitely far apart and the potential of the external field is zero at that point.

A dipole is expressed as a pair of fixed and opposite charges separated by a fixed distance. It is most common in dielectric materials, molecules, and in various electrical systems. The electric dipole experiences both torque and a change in its potential energy when placed in an external field.

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NCERT Class 11 notes	Class 11 Physics NCERT notes
Class 11 Chemistry notes	CBSE Class 11 maths notes

Diple Moment:

The dipole moment is given as: $\stackrel{\to }{p}=q\stackrel{\to }{d}$

Where,

• q is the magnitude of each charge.
• $\bar{d}$ is the displacement vector from the negative to the positive charge

Formula for Potential Energy:

The potential energy for the dipole moment in an external field is given as:

$U=-\bar{p}·\bar{E}$

Here, 

• p is the magnitude of the dipole moment.
• E is the magnitude of the external electric field.

Example 1: If we place a charge 𝑞 = 2𝜇C in a uniform electric field 𝐸⃗ = 1000 N/C𝑖ˆ at position 𝑥 = 0.1 m, how much will be its potential energy? Assume 𝑉 = 0 at 𝑥 = 0.

Solution:

The potential at 𝑥 = 0.1 m :
𝑉 = −𝐸𝑥 = −1000 ⋅ 0.1 = −100 V

Potential energy: $U=qV=(2\times {10}^{-6})·(-100)=-2\times {10}^{-4}\text{J}=-0.0002\text{J}$

The negative value indicates the charge is in a position where work is required to move it against the field.

Example 2: We place a dipole with moment 𝑝 = $5\times {10}^{-10}\text{Cm}$ in a uniform field 𝐸⃗ = 2000 N/C. Calculate the potential energy when the dipole is oriented at 𝜃 = 60 degrees to the field.

Solution:

𝑈 = −𝑝𝐸cos 𝜃

= $-(5\times {10}^{-10})·2000·\cos 60°=-{10}^{-6}·0.5=-5\times {10}^{-7}\text{J}$

 

 

The negative energy tells us that it's a stable configuration, even when it's aligned partially.