Unit 10: Electric Force, Field, and Potential
Weightage : 15–18%
The voltage between the plates of a fully charged parallel-plate capacitor will
Correct Answer: \( \boxed{\mathrm{B}} \)
Between the plates of an ideal parallel-plate capacitor, the electric field is uniform.
The relationship between electric field and electric potential is
\( E=-\dfrac{dV}{dx}. \)
Since the electric field \(E\) is constant, the rate of change of potential with distance is also constant. Therefore, the electric potential decreases uniformly with distance between the plates.
Equivalently,
\( \Delta V=Ed, \)
where \(d\) is the distance measured from one plate. This is a linear relationship, so the potential falls linearly from the positive plate to the negative plate.
Therefore, the correct answer is (B).
In the figure above, equipotential lines are drawn at \(0\ \mathrm{V}\), \(20.0\ \mathrm{V}\), and \(40.0\ \mathrm{V}\). The total work done in moving a point charge of \(+3.00\ \mathrm{mC}\) from position \(a\) to position \(b\) is
Correct Answer: \( \boxed{\mathrm{D}} \)
The work required to move a charge between two points depends only on the potential difference between the points, not on the path taken.
The work done is
\( \displaystyle W=q\Delta V \)
Here,
\( \displaystyle q=3.00\times10^{-3}\ \mathrm{C} \)
\( \displaystyle \Delta V=40.0\ \mathrm{V}-0\ \mathrm{V}=40.0\ \mathrm{V} \)
Therefore,
\( \displaystyle W=\left(3.00\times10^{-3}\right)(40.0)=0.120\ \mathrm{J}=120\ \mathrm{mJ} \)
The vertical portion of the motion is along an equipotential line, so it requires no additional work.
Therefore, the correct answer is (D).
Which of the following relationships, when plotted, will yield a curve that is inverse to the first power?
I. The electric potential versus distance from a positive point particle.
II. The volume versus pressure for an ideal gas.
III. The magnetic field from a current-carrying wire versus distance from the wire.
Correct Answer: \( \boxed{\mathrm{D}} \)
Each of the given relationships is proportional to the inverse of the independent variable.
For a point charge, the electric potential is
\(V=\dfrac{kQ}{r},\)
so \(V\propto\dfrac{1}{r}\).
For an ideal gas with fixed \(n\) and \(T\), the ideal gas law gives
\(PV=nRT,\)
or
\(P=\dfrac{nRT}{V},\)
which means \(P\propto\dfrac{1}{V}\). Equivalently, plotting volume versus pressure also produces an inverse first-power relationship.
For a long straight current-carrying wire, the magnetic field is
\(B=\dfrac{\mu_0 I}{2\pi r},\)
so \(B\propto\dfrac{1}{r}\).
Therefore, all three relationships are inverse to the first power.
Therefore, the correct answer is (D).
