Electric Potential Due to Dipole

Have you ever tried filling up a glass of water to the absolute edge of the top? As you pour smaller and smaller increments of water into the glass, you'll notice that it takes a substantially large volume of water for the water to finally overflow and spill out from the glass. This is due to a phenomenon called surface tension, which allows the water at the top surface of the glass to have tension and reduce to a minimum surface area, thus resulting in a larger amount of water being held at the surface. 

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Two charges \(+q\) and \(-q\) are placed at point \(A=(a,0)\) and \(B=(-a,0)\) on the \(x\)-axis. The potential due to this system of charges at the point \(P=(r,\theta)\) will be

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What is the formula for an electric potential of a dipole when \(a\) is the length of the dipole and \(r\) is the distance of an observation point P from the center of the dipole making an angle \(\theta\) with the length of the dipole?

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The electric potential at a point due to a point charge is directly proportional to the distance of that point to the point charge.

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The electric potential due to a dipole is directly proportional to the magnitude of the charge on either end of the dipole.

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What is an electric potential due to a dipole on an axial line, where the axial line is the line through the axis of the dipole?

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What is an electric potential due to a dipole on an equatorial line? 

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The electric potential due to a dipole is inversely proportional to the square of the distance of the observation point to the center of the dipole.

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The electric potential due to a dipole is _____ proportional to the dipole moment.

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What is the SI unit of electric potential?

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The electric potential due to a dipole is inversely proportional to the length of the dipole.

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What is the direction of the electric dipole moment of a dipole?

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Two charges \(+q\) and \(-q\) are placed at point \(A=(a,0)\) and \(B=(-a,0)\) on the \(x\)-axis. The potential due to this system of charges at the point \(P=(r,\theta)\) will be

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What is the formula for an electric potential of a dipole when \(a\) is the length of the dipole and \(r\) is the distance of an observation point P from the center of the dipole making an angle \(\theta\) with the length of the dipole?

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The electric potential at a point due to a point charge is directly proportional to the distance of that point to the point charge.

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The electric potential due to a dipole is directly proportional to the magnitude of the charge on either end of the dipole.

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What is an electric potential due to a dipole on an axial line, where the axial line is the line through the axis of the dipole?

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What is an electric potential due to a dipole on an equatorial line? 

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The electric potential due to a dipole is inversely proportional to the square of the distance of the observation point to the center of the dipole.

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The electric potential due to a dipole is _____ proportional to the dipole moment.

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What is the SI unit of electric potential?

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The electric potential due to a dipole is inversely proportional to the length of the dipole.

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What is the direction of the electric dipole moment of a dipole?

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    Electric Potential Due to a Dipole Water droplet StudySmarterFig. 1 - The surface tension of water is what allows for the water droplet to keep its shape.

    The phenomenon of water tension is due to the fact that the water molecules making up the liquid are electric dipoles. As water molecules are made up of hydrogen and oxygen, where hydrogen is positively charged and oxygen is negatively charged, the two opposite charges create an electric dipole. This allows for the water molecules to form a strong bond with each other, resulting in surface tension. This article will be focusing on the electric potential due to a dipole, keep reading to find out more!

    Potential Due to a Dipole at Any Point

    Before calculating the electric potential due to an electric dipole at a point, it is essential to understand what is an electric potential and electric dipole individually. So let's start with an electric dipole.

    An electric dipole is a setup consisting of equal and opposite charges separated by a certain distance.

    The measurement of the strength of an electric dipole is given by the dipole moment \(\vec{p}\). It is a vector quantity whose direction is from negative to positive charge along the dipole length.

    Let's consider two point charges \(+q\) and \(-q\) separated by a certain distance say \(2a\).

    Electric Potential Due to a Dipole Electric Dipole StudySmarterFig. 2 - The electric dipole moment of an electric dipole points from a negative charge \(-q\) towards a positive charge \(+q\).

    The magnitude of an electric dipole moment of a dipole shown in the diagram is\[\left|\vec{p}\right|=2qa,\] and its SI unit is the coulomb-meter \(\left(\mathrm{C\,m}\right)\).

    The midpoint O in the diagram represents the center of the dipole. The net charge on the dipole is \(-q+q=0\,\mathrm{C}\). Some examples in which the center of a negative charge is separated from the center of a positive distribution are water, alcohol, and HCL.

    Now, let's learn what an electric potential is and why it is important to know about the electric potential due to a dipole.

    The electric potential of a charged body is the amount of energy required to move a unit of charge from infinity to that point.

    An electric potential determines the direction of the flow of charge. A positive electric charge always accelerates from high potential to low potential. The flow of charge stops as soon as the potential gradient becomes zero.

    Imagine a single-point charge \(+q\) at a point O. The electric potential at the point which is at a distance of \(r\) from the point O is the amount of work done in moving per unit of positive charge to move it from infinity to the point P against the electrostatic force of repulsion:\[V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}.\]

    Potential due to a dipole electric potential StudySmarterFig. 3 - The electrostatic potential at point P tells us how much work you have to do to move a unit positive charge of \(+1\,\mathrm{C}\) from infinity to the point P against the electrostatic force of repulsion due to the point charge \(+q\).

    An infinite point is considered a reference point where the value of the electric potential is zero. The unit positive charge shown in Figure 3 is moving from zero potential to some positive value of potential.

    Everything in the real world is made of matter, and matter contains atoms or molecules which are electrically neutral. A molecule has a positively charged nucleus and negatively charged electrons. If the center of positive charges does not coincide with the center of negative charges, then the molecule behaves as a polar molecule, an electric dipole. The space around which an electric effect of a dipole can be experienced is called the dipole field. When a charge is placed in this space, it experiences an electric force.

    The electric potential due to an electric dipole at any point tells us the amount of work to bring a unit positive charge from a reference point very far away to a specific point against the electric field due to the dipole.

    In the next part, we will derive the formula for an electric potential due to a dipole.

    Electric Potential Due to a Dipole Derivation

    Imagine two equal and unlike charges \(-q\) and \(+q\) separated by a distance \(2a\). The dipole moment of this electric dipole is \(\left|\vec{p}\right|=q2a\). Let P be an observation point where we will calculate the electric potential due to this electric dipole.

    Electric potential due to a dipole Electric potential at any point due to a dipole StudySmarterFig. 4 - The figure shows an observation point P at a distance of \(r\) from the center of a dipole, i.e., point O, such that its position vector makes an angle of \(\theta\) with the length of the dipole.

    From the above diagram,

    1. The position vector of P from the center of the dipole is \(\vec{\text{OP}}=\vec{r}\),

    2. \(\angle \text{BOP}=\theta\),

    3. Distance of P from the ends of the dipole, i.e., A and B, is \(\text{AP}=r_1\) and \(\text{BP}=r_2\).

    Electric Potential at point P due to a charge \(-q\) at point A is \[V_1=-\frac{q}{4\pi\epsilon_0r_1}\tag{1}\] Similarly, electric potential at point P due to a charge \(+q\) at point B is \[V_2=\frac{q}{4\pi\epsilon_0r_2}.\tag{2}\] By using the superposition principle, the electric potential at point P due to both the charges at points A and B is \[V=V_1+V_2.\] Substituting the values from equations (1) and (2) in the above equation, \[\begin{align*}V&=-\frac{q}{4\pi\epsilon_0r_1}+\frac{q}{4\pi\epsilon_0r_2}\\V&=\frac{q}{4\pi\epsilon_0}{\left(-\frac{1}{r_1}+\frac{1}{r_2}\right)}\\V&=\frac{q}{4\pi\epsilon_0}{\left(\frac{1}{r_2}-\frac{1}{r_1}\right)}.\tag{3}\end{align*}\]

    This equation gives the value of electric potential due to a dipole. In the next part, we will get the formula of electric potential due to a dipole which can be used in axial (when the observation point is on the line parallel to the length of the dipole) and equatorial case(when the observation is on the line perpendicular to the length of the dipole).

    The remainder of this definition requires knowledge of a Taylor expansion which is only covered in AP Calculus.

    In Figure 4, the position vector of P from point O makes an angle \(\theta\) with the length of the dipole. Using the triangle law of vector addition \[r_1^2=r^2+a^2+2ar\cos{\left(\theta\right)}\tag{4}\] and \[\begin{align*}r_2^2&=r^2+a^2+2ar\cos{\left(180^\circ-\theta\right)}\\r_2^2&=r^2+a^2-2ar\cos{\left(\theta\right)}\tag{5}\end{align*}\]

    Equation (4) can also be written as \[r_1^2=r^2\left(1+\frac{a^2}{r^2}+2\frac{a}{r}\cos{\left(\theta\right)}\right)\]

    Let's assume \(a\ll r\), such that \(\frac{a}{r}\) is small and \(\frac{a^2}{r^2}\) can be neglected. \[ r_1^2=r^2\left(1+2\frac{a}{r}\cos{\left(\theta\right)}\right)\] or \[\begin{align*}r_1&=r\left(1+2\frac{a}{r}\cos{\left(\theta\right)}\right)^{1/2}\\\frac{1}{r_1}&=\frac{1}{r}\left(1+2\frac{a}{r}\cos{\left(\theta\right)}\right)^{-1/2}\end{align*}\]

    With the Taylor expansion and neglecting the higher power of \(\frac{a}{r}\) we get \[\begin{align*}\frac{1}{r_1}&=\frac{1}{r}\left[1+2\frac{a}{r}\cos{\left(\theta\right)}\times\left(-\frac{1}{2}\right)\right]\\\frac{1}{r_1}&=\frac{1}{r}\left[1-\frac{a}{r}\cos{\left(\theta\right)}\right]\end{align*}\tag{6}\]

    Similarly, equation (5) becomes \[\frac{1}{r_2}=\frac{1}{r}\left[1+\frac{a}{r}\cos{\left(\theta\right)}\right]\tag{7}\]

    Using equations (3), (6), and (7) \[\begin{align*}V&=\frac{q}{4\pi\epsilon_0}\left[\frac{1}{r}\left(1+\frac{a}{r}\cos{\left(\theta\right)}\right)-\frac{1}{r}\left(1-\frac{a}{r}\cos{\left(\theta\right)}\right)\right]\\V&=\frac{q}{4\pi\epsilon_0r}\left[1+\frac{a}{r}\cos{\left(\theta\right)}-1+\frac{a}{r}\cos{\left(\theta\right)}\right]\\V&=\frac{q 2a}{4\pi\epsilon_0r^2}\cos{\left(\theta\right)}\end{align*}\]

    or \[V=\frac{p}{4\pi\epsilon_0r^2}\cos{\left(\theta\right)}\] where \(p\) is an electric dipole moment.

    Therefore, the formula of electric potential due to a dipole at any point P is \[V=\frac{p}{4\pi\epsilon_0r^2}\cos{\left(\theta\right)}\tag{9}\]

    Electric Potential due to a Dipole Formula

    Now we have established our equation for the electric potential of a dipole as

    \[ V = \frac{p}{4 \pi \epsilon_0 r^2}\cos{\left(\theta\right)},\]

    where \(p\) is the electric dipole moment measured in \(\mathrm{C \, m}\), \(V\) is the electric potential measured in units of volts \(\mathrm{V}\), \(\epsilon_0\) is the permittivity of free-space with a value of \( 8.85 \times 10^{-12} \, \mathrm{\frac{F}{m}}\), and \(\theta\) is the angle between the point and the dipole measured in \(\mathrm{rads}\).

    Electric Potential due to a Dipole on an Axial Line

    Imagine an observation point P on the line, passing through both ends of the dipole.

    Electric Potential due to a dipole on axial line StudySmarterFig. 5 - Observation point P on the line passing through both the ends of a dipole such that an angle between the position vector of P from the center of a dipole and the length of the dipole is \(0^\circ\).

    As the angle between OP and AB is \(0^\circ\), thus, the electric potential due to a dipole at point P, in this case, is \[\begin{align*}V_{\mathrm{axial}}&=\frac{p}{4\pi\epsilon_0r^2}\cos{0^\circ}\\V_{\mathrm{axial}}&=\frac{p}{4\pi\epsilon_0r^2}\end{align*}\] or \[V_{\mathrm{axial}}=\frac{q2a}{4\pi\epsilon_0r^2}.\] This above equation gives an electric potential due to a dipole on the axial line.

    The equation shows that:

    1. the electric potential due to a dipole is maximum on the axial line,

    2. the electric potential varies inversely with the square of the distance from an observation point to the center of the dipole,

    3. the electric potential varies directly with the length of the dipole,

    4. the electric potential varies directly with the strength of the charge on either side of the dipole.

    Electric Potential due to a Dipole on an Equatorial Line

    Imagine an observation point P is on the line perpendicular to the length of the dipole.

    Electric potential due to a dipole on equatorial line StudySmarterFig. 6 - Observation point P on the line perpendicular to the length of an electric dipole such that the angle between its position vector from the center of the dipole and the length of the dipole is \(90^\circ\)

    From the above Figure, the angle between OP and AB is \(90^\circ\). Thus, the electric potential due to a dipole at point P, in this case, is \[\begin{align*}V_{\mathrm{eq}}&=\frac{p}{4\pi\epsilon_0r^2}\cos{90^\circ}\\V_{\mathrm{eq}}&=0\end{align*}\] The above equation shows that an electric potential due to a dipole on an equatorial line is zero. This is logical because the distance to either charge is the same on this line, so the point experiences equal but opposite voltages due to these charges.

    Electric Potential Due to Dipole - Key takeaways

    • An electric dipole is a setup consisting of equal and unlike charges separated by a certain distance.
    • An electric potential of a charged body represents a degree of electrification of a body.
    • Electric potential due to an electric dipole at any point is the amount of work done in bringing a unit positive charge from a reference point to a specific point against the electric field due to a dipole.
    • Electric potential due to a dipole at any point P is \(V=\frac{q 2a}{4\pi\epsilon_0r^2}\cos{\left(\theta\right)}\) where \(r\) is the distance of point P from the center of a dipole, \(2a\) is the length of a dipole, \(q\) is the magnitude of the charge on either end of the dipole, and \(\theta\) is an angle between the position vector of point P from the center of dipole and the length of a dipole.
    • Electric potential due to a dipole on an axial point is \(V_{\mathrm{axial}}=\frac{q\times 2a}{4\pi\epsilon_0r^2}\), which is the maximum value of electric potential due to a dipole.
    • Electric potential due to a dipole on an equatorial line is \(V_{\mathrm{eq}}=0\) which is the minimum value of electric potential due to a dipole.

    References

    1. Fig. 1 - Water droplet, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Dew,_surface_tension_01.jpg) Licensed by CC BY-SA 4.0 ( https://creativecommons.org/licenses/by-sa/4.0/)
    2. Fig. 2 - Electric dipole moment of an electric dipole, StudySmarter Originals.
    3. Fig. 3 - Electric potential at a point in terms of work done, StudySmarter Originals.
    4. Fig. 4 - Electric potential due to a dipole at any point, StudySmarter Originals.
    5. Fig. 5 - Electric potential due to a dipole on axial line, StudySmarter Originals.
    6. Fig. 6 - Electric Potential due to a dipole on an equatorial line, StudySmarter Originals.
    Frequently Asked Questions about Electric Potential Due to Dipole

    On what factors does an electric potential due to a dipole depends?

    The formula of an electric potential due to a dipole at any point depends upon,

    1. the length of a dipole
    2. the distance of an observation point from the center of a dipole
    3. an angle between the position vector and the length of a dipole
    4. the magnitude of the charge on either end of a dipole.

    What is the electrostatic potential due to an electric dipole? 

    Electric potential due to an electric dipole at any point is the amount of work done in bringing a unit positive charge from a reference point to a specific point against the electric field due to a dipole.

    What is an electric potential due to a dipole on an equatorial line? 

    The electric potential due to a dipole on an equatorial line is zero.

    Is electric potential zero at the midpoint of a dipole? 

    Yes, due to the canceling of electric potential due to either charge, the net electric potential at the midpoint of a dipole is zero.

    How is the potential due to a dipole derived? 

    We can derive the potential due to a dipole using the superposition principle and a formula of electric potential due to a single-point charge.

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    Test your knowledge with multiple choice flashcards

    Two charges \(+q\) and \(-q\) are placed at point \(A=(a,0)\) and \(B=(-a,0)\) on the \(x\)-axis. The potential due to this system of charges at the point \(P=(r,\theta)\) will be

    What is the formula for an electric potential of a dipole when \(a\) is the length of the dipole and \(r\) is the distance of an observation point P from the center of the dipole making an angle \(\theta\) with the length of the dipole?

    The electric potential at a point due to a point charge is directly proportional to the distance of that point to the point charge.

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