Electrostatic Potential

What the chapter starts from

• The Coulomb force, like gravity, is conservative: the work done in moving a charge from one point to another is independent of the path, and zero around any closed loop. Only because of this can a potential energy be defined at all.
• Bring a test charge $q$ from $R$ to $P$ against the electrostatic force, applying an external force $\vec F_{\text{ext}}=-\vec F_E$ at every point so there is no acceleration. The work done is stored as potential energy.

$$\Delta U=U_P-U_R=W_{RP}$$

• Two riders CBSE expects you to state. First, the test charge must be small enough not to disturb the source configuration. Second, $\Delta U$ depends only on the end points, which is exactly the statement that the force is conservative.
• Potential energy is undetermined to within an additive constant: replacing $U$ by $U+\alpha$ everywhere leaves every difference unchanged. The conventional choice is $U=0$ at infinity, which gives $U_P=W_{\infty P}$.

Definition of electrostatic potential

Because the force on a test charge is $q\vec E$, the work is proportional to $q$. Dividing it out leaves a quantity that belongs to the field alone.

$$V_P-V_R=\dfrac{U_P-U_R}{q}=\dfrac{W_{RP}}{q}$$

• Potential difference between two points = work done per unit positive charge by an external agent in moving it from one point to the other, without acceleration.
• Taking $V=0$ at infinity gives the definition to quote in a 1- or 2-mark answer.

$$V=\dfrac{W_{\infty\to P}}{q}$$

• Electrostatic potential at a point is the work done by an external force in bringing a unit positive charge from infinity to that point, without acceleration.
• Strictly, take an infinitesimal test charge $dq$, find $dW$, and use $V=\dfrac{dW}{dq}$, so that the test charge never disturbs the sources.

Units, dimensions and sign

• $V$ is a scalar. SI unit the volt: $1\ \text{V}=1\ \text{J C}^{-1}$. Since $E=\dfrac{V}{d}$, the volt can also be written $1\ \text{N C}^{-1}\text{m}$ — this is the answer to the 1-mark question asking which quantity has the unit $\text{N C}^{-1}\text{m}$.
• Dimensional formula $[\text{M}^{1}\text{L}^{2}\text{T}^{-3}\text{A}^{-1}]$.
• Potential at a point is positive near a positive charge, negative near a negative charge. Only differences are physically significant.
• The electron volt: an electron accelerated through $\Delta V=1\ \text{V}$ gains $e\Delta V=1.6\times10^{-19}\ \text{J}$, defined as $1\ \text{eV}$. Then $1\ \text{keV}=1.6\times10^{-16}\ \text{J}$, $1\ \text{MeV}=1.6\times10^{-13}\ \text{J}$, $1\ \text{GeV}=1.6\times10^{-10}\ \text{J}$.

Work done by the agent and by the field

$$W_{\text{ext}}=q(V_B-V_A),\qquad W_{\text{field}}=-W_{\text{ext}}=q(V_A-V_B)$$

• Moving a positive charge from low to high potential requires positive external work; the field then does negative work.
• Around any closed path, $\oint\vec E\cdot d\vec l=0$, so the work done in taking a charge around a closed loop is zero. That single line is the whole answer to the board question about a charge $q$ carried round a closed rectangular loop $abcda$ between capacitor plates: the electrostatic field is conservative, hence $W=0$.

Diagram to describe in words

Draw a positive source charge $Q$ at the origin $O$, and two points $R$ (far) and $P$ (near) on a radial line. Mark the repulsive electric force $\vec F_E$ on the test charge $q$ pointing away from $O$, and the applied external force $\vec F_{\text{ext}}$ of equal magnitude pointing towards $O$. Label the displacement from $R$ to $P$. A second sketch shows two different curved paths from $R$ to $P$, annotated "work is the same along both", to make the conservative-force point visually.

Standard results