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.
Explore our app and discover over 50 million learning materials for free.
Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken
Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.
Jetzt kostenlos anmeldenHave 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.
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!
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 \(2a\).
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:\[V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}.\]
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.
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.
From the above diagram,
The position vector of P from the center of the dipole is \(\vec{\text{OP}}=\vec{r}\),
\(\angle \text{BOP}=\theta\),
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 \[V=\frac{p}{4\pi\epsilon_0r^2}\cos{\left(\theta\right)}\tag{9}\]
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 \(\mathrm{rads}\).
Imagine an observation point P on the line, passing through both ends of the dipole.
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:
the electric potential due to a dipole is maximum on the axial line,
the electric potential varies inversely with the square of the distance from an observation point to the center of the dipole,
the electric potential varies directly with the length of the dipole,
the electric potential varies directly with the strength of the charge on either side of the dipole.
Imagine an observation point P is on the line perpendicular to the length of the dipole.
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.
The formula of an electric potential due to a dipole at any point depends upon,
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.
The electric potential due to a dipole on an equatorial line is zero.
Yes, due to the canceling of electric potential due to either charge, the net electric potential at the midpoint of a dipole is zero.
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.
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?
\(V=\frac{qa}{4\pi\epsilon_0r^2}\cos{\theta}\).
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?
\(V=\frac{q2a}{4\pi\epsilon_0r^2}\).
What is an electric potential due to a dipole on an equatorial line?
0
The electric potential at a point due to a point charge is directly proportional to the distance of that point to the point charge.
False.
The electric potential due to a dipole is directly proportional to the magnitude of the charge on either end of the dipole.
True.
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.
True.
Already have an account? Log in
Open in AppThe first learning app that truly has everything you need to ace your exams in one place
Sign up to highlight and take notes. It’s 100% free.
Save explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.
Already have an account? Log in
Already have an account? Log in
The first learning app that truly has everything you need to ace your exams in one place
Already have an account? Log in