Elecrostatics flash cards

1. 1.State Coulomb's law in vector form for the force on charge $q_1$ due to $q_2$.

   $\vec{F}_{12}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}\,\hat{r}_{21}$, where $\hat{r}_{21}$ points from $q_2$ to $q_1$. Like charges repel, unlike attract.

   Hint: Inverse-square, along the line joining the charges.

2. 2.What is the value of $\dfrac{1}{4\pi\varepsilon_0}$ in SI units?

   $\dfrac{1}{4\pi\varepsilon_0}\approx 9\times10^{9}\ \text{N m}^2/\text{C}^2$, with $\varepsilon_0=8.85\times10^{-12}\ \text{C}^2/\text{N m}^2$.

   Hint: Coulomb constant $k$.

3. 3.How does the electrostatic force between two point charges change if a dielectric of constant $K$ fills the space?

   The force reduces by a factor $K$: $F_{medium}=\dfrac{F_{vacuum}}{K}=\dfrac{1}{4\pi\varepsilon_0 K}\dfrac{q_1q_2}{r^2}$.

   Hint: Replace $\varepsilon_0$ with $\varepsilon=K\varepsilon_0$.

4. 4.State the principle of superposition for electrostatic forces.

   The net force on a charge is the vector sum of the individual Coulomb forces from every other charge, each computed as if the others were absent.

   Hint: Forces add as vectors, independently.

5. 5.Define the electric field $\vec{E}$ at a point.

   $\vec{E}=\displaystyle\lim_{q_0\to0}\dfrac{\vec{F}}{q_0}$ — force per unit positive test charge. Units: $\text{N/C}$ or $\text{V/m}$.

   Hint: Test charge taken small so it doesn't disturb the source.

6. 6.Electric field due to a point charge $q$ at distance $r$?

   $\vec{E}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}\,\hat{r}$. Points radially outward for $q>0$, inward for $q<0$.

   Hint: Same inverse-square shape as Coulomb force.

7. 7.List key properties of electric field lines.

   Start on $+$ charges, end on $-$; never cross; tangent gives $\vec{E}$ direction; density $\propto$ field magnitude; no closed loops in electrostatics; perpendicular to conductor surfaces.

   Hint: Density and direction encode the field.

8. 8.Why can two electric field lines never intersect?

   At an intersection the field would have two directions at once, which is impossible since $\vec{E}$ is uniquely defined at each point.

   Hint: Tangent = field direction, must be single-valued.

9. 9.Define electric dipole moment $\vec{p}$.

   $\vec{p}=q\,\vec{d}$ (magnitude $q\times 2a$ for separation $2a$), directed from the negative to the positive charge. Units: $\text{C\,m}$.

   Hint: Points from $-q$ to $+q$.

10. 10.Field on the axial line of a dipole at distance $r$ ($r\gg a$)?

    $E_{axial}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$, directed parallel to $\vec{p}$.

    Hint: Twice the equatorial value, $1/r^3$.

11. 11.Field on the equatorial line of a dipole at distance $r$ ($r\gg a$)?

    $E_{eq}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}$, directed antiparallel to $\vec{p}$.

    Hint: Half the axial value, opposite direction.

12. 12.How does a dipole's field fall off with distance compared to a point charge?

    Dipole field $\propto 1/r^3$, while a point charge field $\propto 1/r^2$. The dipole falls faster.

    Hint: Net charge zero speeds the decay.

13. 13.Torque on a dipole $\vec{p}$ in a uniform field $\vec{E}$?

    $\vec{\tau}=\vec{p}\times\vec{E}$, magnitude $\tau=pE\sin\theta$. It tends to align $\vec{p}$ with $\vec{E}$.

    Hint: Cross product; zero when aligned or anti-aligned.

14. 14.Net force on a dipole in a uniform field?

    Zero — the forces $+q\vec{E}$ and $-q\vec{E}$ are equal and opposite. Only a torque acts (unless the field is non-uniform).

    Hint: Equal & opposite forces on the two ends.

15. 15.Potential energy of a dipole in a uniform field?

    $U=-\vec{p}\cdot\vec{E}=-pE\cos\theta$. Minimum ($-pE$) when aligned, maximum ($+pE$) when anti-aligned.

    Hint: Zero at $\theta=90^\circ$ by convention.

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