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NCERT Exemplar · Class 12 Physics Electrostatic Potential and Capacitance

33 questions

Very Short Answer Questions 14–18 (part 3 of 5)

  1. Exercise 14

    Consider two conducting spheres of radii R1\displaystyle R_1 and R2\displaystyle R_2 with R1>R2\displaystyle R_1>R_2. If the two are at the same potential, the larger sphere has more charge than the smaller sphere. State whether the charge density of the smaller sphere is more or less than that of the larger one.
    NCERT’s answer
    More.
    Same potential \(\displaystyle V\) on both spheres:\[V=\frac{Q}{4\pi\varepsilon_0 R}\;\Rightarrow\;Q=4\pi\varepsilon_0 R\,V \]\[\sigma=\frac{Q}{4\pi R^2}=\frac{\varepsilon_0 V}{R} \]\[R_2<R_1\;\Rightarrow\;\sigma_2=\frac{\varepsilon_0 V}{R_2}>\frac{\varepsilon_0 V}{R_1}=\sigma_1 \]Answer: The smaller sphere has the greater charge density.

  2. Exercise 15

    Do free electrons travel to region of higher potential or lower potential?
    NCERT’s answer
    Higher potential.
    An electron's force is opposite to \(\displaystyle \mathbf{E}\), and \(\displaystyle \mathbf{E}\) points from high to low potential:\[\mathbf{F}=-e\,\mathbf{E} \]Moving to higher potential lowers its potential energy:\[\Delta U=(-e)\,\Delta V<0\quad\text{for}\quad\Delta V>0 \]Answer: Free electrons travel to the region of higher potential.

  3. Exercise 16

    Can there be a potential difference between two adjacent conductors carrying the same charge?
    NCERT’s answer
    Yes, if the sizes are different.
    Equal charge on conductors of different size gives different potentials. Take adjacent spheres, \(\displaystyle R_1\neq R_2\), centres \(\displaystyle d\) apart (\(\displaystyle d\gg R_1,R_2\)); each also sees its neighbour's charge:\[V_1=\frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{R_1}+\frac{1}{d}\right),\qquad V_2=\frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{R_2}+\frac{1}{d}\right) \]\[\Delta V=V_2-V_1=\frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{R_2}-\frac{1}{R_1}\right)\neq 0 \]Answer: Yes, if their sizes or shapes differ.

  4. Exercise 17

    Can the potential function have a maximum or minimum in free space?
    NCERT’s answer
    No.
    At a maximum (minimum) of \(\displaystyle V\), \(\displaystyle \mathbf{E}=-\nabla V\) points outward (inward) all round, so over a small sphere about that point:\[\oint\mathbf{E}\cdot d\mathbf{A}\neq 0 \]Free space has no charge:\[\oint\mathbf{E}\cdot d\mathbf{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}=0 \]Answer: No.

  5. Exercise 18

    A test charge q\displaystyle q is made to move in the electric field of a point charge Q along two different closed paths (Fig. 2.6\displaystyle 2.6). First path has sections along and perpendicular to lines of electric field. Second path is a rectangular loop of the same area as the first loop. How does the work done compare in the two cases? NCERT_Question_Class12_Physics_Exemplar_Ch2_Q18
    NCERT’s answer
    As electric field is conservative, work done will be zero in both the cases.
    The electrostatic force is conservative, so over any closed loop:\[W=q\oint\mathbf{E}\cdot d\mathbf{l}=0 \]In the first loop the arcs (radii \(\displaystyle r_1\) inner, \(\displaystyle r_2\) outer) are perpendicular to \(\displaystyle \mathbf{E}\), and the radial legs cancel:\[W_1=0+q\,[V(r_1)-V(r_2)]+0+q\,[V(r_2)-V(r_1)]=0 \]Answer: \(\displaystyle W_1=W_2=0\).