Electricity and Magnetism

Charges and Their Interactions

Charges and Their Interactions: Level 3-4 Challenges

         

A particle of mass m=1 kgm=1\text{ kg} and charge q=3 Cq=3\text{ C} is hung from a light inextensible string of length l=2 ml=2\text{ m}. The entire system is placed in a uniform horizontal electric field of magnitude 8 V/m8\text{ V/m}. What must be the minimum initial horizontal velocity with which the particle must be projected so that it completes a vertical circle?

Details and Assumptions

  • The gravitational acceleration is g=10 m/s2g=-10\text{ m/s}^2.
  • The particle is initially at the lowest point.

Consider two parallel, grounded, conducting plates located at x=Lx=-L and x=Lx=L. A charge of Q=92 μCQ=-92 \space \mu\text{C} is placed at (L/2,0,0)(L/2,0,0). What is the total charge induced on the plate located at x=Lx=L in microcoulombs?

Let a charge Q\displaystyle Q be held fixed in space. Another charge q\displaystyle q of mass mm is thrown with a velocity of v\displaystyle v from an infinitely far place, as shown in the figure, such that it is at a separation of d\displaystyle d initially.

Find the minimum distance (in m) between the two charges.

Details and Assumptions

\bullet Q=5 μC\displaystyle Q = 5\ \mu C
\bullet q=1 μC\displaystyle q = 1\ \mu C
\bullet d=3 cm\displaystyle d = 3\text{ cm}
\bullet v=2 m/sec\displaystyle v = 2 \text{ m/sec}
\bullet m=1 g\displaystyle m = 1\text{ g}

The figure shows a non-conducting, non-uniformly charged ring with linear charge densities λ\lambda and λ- \lambda, mass mm and radius RR. This ring is placed on a rough horizontal surface and a horizontal electric field E\vec{E} is switched on.

If at some instant, the ring is in the position shown above and is rolling without slipping, find the minimum coefficient of friction μmin\mu_{min} required.

Details & Assumptions

  1. A uniform gravitational field g=10ms2|\vec{g}| = 10 ms^{-2} acts downward.
  2. λ=π100Cm1\lambda = \dfrac{\pi}{100} Cm^{-1}
  3. m=1.5kgm = 1.5 kg
  4. R=0.15mR = 0.15 m
  5. E=1000NC1|\vec{E}| = 1000 NC^{-1}

A thin ring of radius R\displaystyle R is charged with a charge q\displaystyle q. It is then placed over an infinitely large grounded conducting plane at a height h\displaystyle h, as shown in the figure above.

Find the magnitude of the surface charge density σ\displaystyle \sigma (in C/m2^2) of the induced charge at the point on the plane which is exactly below the ring's center.

Details and Assumptions

\bullet q=2 μC\displaystyle q = 2\ \mu\text{C}
\bullet R=10 cm\displaystyle R = 10\text{ cm}
\bullet h=15 cm\displaystyle h = 15\text{ cm}

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