Electric Potential, Voltage, and Capacitance (4C) - Chem & Physics

1. How does inserting a dielectric with constant $\kappa$ change parallel-plate capacitance?

   $C=\kappa\frac{\varepsilon_0A}{d}$. Dielectric insertion reduces the effective field, increasing capacitance by the factor $\kappa$.

2. What is the SI unit of capacitance?

   $1\ \text{F}=1\ \text{C/V}$. The farad measures capacitance as one coulomb of charge stored per volt of potential difference.

3. State the relation between electric field and electric potential in one dimension.

   $E=-\frac{dV}{dx}$. The electric field is the negative rate of change of potential with respect to position.

4. If $V(r)=\frac{kQ}{r}$, what is the ratio $\frac{V(2r)}{V(r)}$?

   $\frac{V(2r)}{V(r)}=\frac{1}{2}$. Potential inversely proportional to distance, so doubling $r$ halves the potential.

5. Find $Q$ stored on a capacitor with $C=5\ \mu\text{F}$ and $V=12\ \text{V}$.

   $Q=CV=60\ \mu\text{C}$. Charge stored is the product of capacitance and applied voltage.

6. Identify $C_{\text{eq}}$ for two identical capacitors $C$ in series.

   $C_{\text{eq}}=\frac{C}{2}$. For identical capacitors in series, equivalent is half due to doubled effective separation.

7. State the formula for potential difference between two points in terms of work.

   $\Delta V=-\frac{W_{\text{field}}}{q}$. Potential difference equals the negative work done by the field per unit charge when moving a charge between points.

8. Which has zero work by the electric field: motion along or perpendicular to an equipotential surface?

   Perpendicular motion has $W=0$ (along equipotential). Motion perpendicular to the field lines follows equipotential surfaces, resulting in no change in potential and zero work.

9. What is the electric potential energy of two point charges $q_1$ and $q_2$ separated by $r$?

   $U=\frac{kq_1q_2}{r}$. Represents the work to assemble the charges from infinity, analogous to gravitational potential energy.

10. What is the definition of electric potential $V$ at a point?

    $V=\frac{U}{q}$ (electric potential energy per unit charge). Electric potential at a point is the electric potential energy per unit charge for a test charge placed there.

11. What is the SI unit of electric potential (voltage)?

    $1\ \text{V}=1\ \text{J/C}$. The volt is defined as the potential difference that imparts one joule of energy to one coulomb of charge.

12. What is the definition of capacitance $C$ in terms of charge and voltage?

    $C=\frac{Q}{\Delta V}$. Capacitance quantifies the ability to store charge for a given potential difference across the device.

13. State the energy stored in a capacitor in terms of $Q$ and $C$.

    $U=\frac{Q^2}{2C}$. Alternative form obtained by substituting $Q=CV$ into the energy expression.

14. What is the superposition rule for electric potential from multiple charges?

    $V_{\text{net}}=\sum_i \frac{kQ_i}{r_i}$. Electric potential is a scalar quantity, allowing direct summation of individual contributions.

15. State the equivalent capacitance for capacitors in series.

    $\frac{1}{C_{\text{eq}}}=\sum_i \frac{1}{C_i}$. In series, total voltage divides across capacitors, leading to reciprocal sum for equivalent capacitance.

16. State the equivalent capacitance for capacitors in parallel.

    $C_{\text{eq}}=\sum_i C_i$. In parallel, charges add while sharing the same voltage, summing individual capacitances.

17. Identify $C_{\text{eq}}$ for two identical capacitors $C$ in parallel.

    $C_{\text{eq}}=2C$. Parallel connection doubles effective plate area for identical capacitors, doubling capacitance.

18. State the relationship between change in electric potential energy and voltage.

    $\Delta U=q\Delta V$. Change in potential energy equals charge times the potential difference experienced by the charge.

19. Which quantity is continuous across a conductor's surface in electrostatic equilibrium: $V$ or $E$?

    $V$ is constant throughout the conductor. In electrostatic equilibrium, the potential is uniform inside and on the surface of a conductor due to zero internal field.

20. What is the electric potential due to a point charge $Q$ at distance $r$?

    $V=\frac{kQ}{r}$. Derived from integrating the electric field from infinity to $r$, assuming zero potential at infinity.

21. State the energy stored in a capacitor in terms of $C$ and $V$.

    $U=\frac{1}{2}CV^2$. Energy stored derives from the work to charge the capacitor, integrating $Q dV$ from 0 to $V$.

22. In a uniform electric field, what is the relation between $\Delta V$, $E$, and displacement $\Delta x$ along the field?

    $\Delta V=-E\Delta x$. In a uniform field, potential decreases linearly in the direction of the field by the product of field strength and distance.

23. State the capacitance of a parallel-plate capacitor with plate area $A$ and separation $d$ (vacuum).

    $C=\frac{\varepsilon_0A}{d}$. For parallel plates, capacitance increases with area and decreases with separation, proportional to permittivity.

24. Find the energy stored when $C=2\ \mu\text{F}$ and $V=3\ \text{V}$.

    $U=\frac{1}{2}CV^2=9\ \mu\text{J}$. Energy calculation uses the formula derived from integrating work done during charging.

25. What is the sign of $V$ at a point due to a negative source charge $Q<0$?

    $V<0$. For $Q<0$, potential is negative relative to zero at infinity, indicating attractive interaction for positive test charges.

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