Electric Charges and Fields Set-1

Q1. Two point charges $+4.0\ \mu\text{C}$ and $+1.0\ \mu\text{C}$ are fixed on the x-axis at $x=0$ and $x=3.0\ \text{m}$ respectively. At what point (value of $x$ measured from the origin) on the line joining them is the resultant electric field zero?

Q2. Assertion (A): The electric field at every point inside a hollow metallic sphere in electrostatic equilibrium is zero. Reason (R): Excess charge on an isolated conductor in electrostatic equilibrium resides only on its outer surface.

Q3. A uniformly charged thin ring of radius $R=0.10\ \text{m}$ carries total charge $Q=5.0\times10^{-6}\ \text{C}$. The axial field is $E(x)=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Qx}{(x^2+R^2)^{3/2}}$. At what magnitude does the axial electric field attain its maximum value?

Q4. A solid non-conducting sphere of radius $R=0.10\ \text{m}$ carries total charge $Q=8.0\times10^{-6}\ \text{C}$ uniformly distributed in its volume. What is the magnitude of the electric field at a point $r=0.05\ \text{m}$ from the centre?

Q5. Assertion (A): The electric potential is continuous across a surface that carries surface charge (no finite jump in $V$). Reason (R): The normal component of the electric field across a surface carrying surface charge density $\sigma$ has a discontinuity given by $\dfrac{\sigma}{\varepsilon_0}$.

Q6. A point charge $q=2.0\times10^{-6}\ \text{C}$ is held at a distance $d=5.0\times10^{-2}\ \text{m}$ from an infinite grounded conducting plane. Using the method of images, what is the magnitude and direction of the electrostatic force on $q$?

Q7. Two point charges $+Q$ and $-Q$ with $Q=2.0\times10^{-6}\ \text{C}$ are located at $x=-a$ and $x=+a$ respectively with $2a=0.10\ \text{m}$. At the midpoint $x=0$, what are the electric potential and the electric field (magnitude and direction)?

Q8. Assertion (A): If the electric potential at a point is zero, then the electric field at that point must be zero. Reason (R): A zero value of potential at a point implies all spatial derivatives of the potential at that point are zero.

Q9. Three qualitative curves of electric field magnitude $E$ versus radial distance $r$ from the origin are described:

Curve (A): $E\to\infty$ as $r\to0$ and for large $r$ behaves as $E\propto 1/r^{2}$.

Curve (B): $E=0$ at $r=0$, rises to a maximum at $r=R/\sqrt{2}$, then for large $r$ behaves as $E\propto 1/r^{2}$.

Curve (C): $E\to\infty$ as $r\to0$ and for large $r$ behaves as $E\propto 1/r$.

Which assignment of physical charge-distributions to curves (A,B,C) is correct?

Q10. Experimental data for a spherically symmetric charge distribution show: for $r<a$, $E(r)\propto r$; for $r>a$, $E(r)\propto 1/r^{2}$. Which charge distribution best matches these measurements?