Chapter#02: Electric Potential

Team Quanta gladly presents all possible short questions of Electricity & Magnetism – I’s Chapter#02: Electric Potential.

Q.1 Distinguish between potential difference and potential energy?

Answer:

Q.2 Give a comparison of a potential due to point charge, dipole, and quadrupole.

Answer: Potential is defined as work done on a unit charge in moving it from an infinite point to any specific point in the electric field. For point charge $V=\frac{kq}{r}$. For a dipole $V=kq\left[\frac{1}{r_+}-\frac{1}{r_-}\right]$ and for quadrupole $V=kq\left[\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}\right]$

Q.3 What is work done in moving a positive charge on equipotential surface?

Answer: The work done on a unit charge moving it along the equipotential surface is zero, because there is equal potential across the equipotential.

$$W =q \Delta V$$

$$W=q\left(0\right)$$

$$W=0$$

Q.4 How can one ensure that electric potential in a given region of space has a constant value?

Answer: The electric potential in a region of space will be constant if the electric field $\vec{E}$ is equal to zero. Mathematically:

$$\vec{E}=0$$

If V is constant

$$\vec{E}=0$$

Q.5 If E equals zero at a given point, must V equals zero for that point? Give an example to prove your answer.

Answer: No, if electric field at a given point is equal to zero the potential should not be necessarily equal to zero. Rather it could be a non-zero constant. As the electric field is negative gradient of potential, so electric field only becomes zero where potential is constant with respect to position.

$$\vec{E}=-\vec{\nabla}.V$$

Q.6 What is an electron volt. Find its value in joules.

Answer: Electron volt is the unit of energy. And is defined as the amount of energy required to move it from one point to another having potential difference of 1 V.

$$1\ eV=1.6\times{10}^{-19}\ CV$$

As in SI units 1 V = 1 J/C

$$1\ eV=1.6\times{10}^{-19}\ C\ J/C$$

$$1\ eV=1.6\times{10}^{-19}\ J$$

Q.7 Why can an isolated atom not have a permanent electric dipole moment?

This is because the electrons are aligned in an atom in such way that they become symmetric. So, for a symmetric distribution of charges, all the dipole moments present inside the atoms will cancel their effect. Hence, net moment of the atom is equal to zero.

Q.8 Explain why equipotential surfaces are always perpendicular to electric field lines?

Answer: No work is done in moving a charge from one point on equipotential surface to the other. Therefore, component of electric field intensity along the equipotential surface is zero it means, the electric field intensity is perpendicular to equipotential surface. Hence, the surface is perpendicular to field lines.

Q.9 A proton is released from rest in uniform electric field. What happens to its electric potential energy?

Answer: When a proton is released from rest in a uniform electric field it tends to move from higher potential to lower potential. Hence, its overall potential energy decreases all the way when it moves along field lines.

Q.10 What would happen to you if you were on an insulated stand and your potential was increased by 10 kV with respect to the earth?

Answer: The current will not flow though you when you were on the insulated stand and your potential was increased by 10 kV with respect to the Earth because circuit is not completed.

Q.11 Would a proton volt compare with an electron volt? The mass of a proton is1840 times that of an electron?

Answer: It makes no difference either we use proton volt or electron volt because electron volt = eV = charge of electron x 1 V. The charge of electron and charge of proton is same in magnitude.

Q.12 Do electrons tend to go to region of high potential or of low potential?

Answer: The electron tends to go from low potential region to  high  potential region. The low potential (-) side pushes negatively charged electron and high potential (+) side attracts negatively charged electron.

Q.13 Does the amount of work per unit charge required to transfer electric charge from one point to another point in electric field depends upon the amount of charge transferred?

Answer: We know that V = W/q. Therefore, the amount of work per unit charge required to transfer electric charge from one point to another point in electric field depends upon the amount of charge transferred.

Q.14 Can there be a potential difference between two conductors that carry like charges of the same magnitude?

Answer: There will be no potential difference between two conductors that carry like charges of the same magnitude.

Q.15 We have seen that inside a hollow conductor, you are shielded from the fields of outside charges. Why?

Answer: The shell theorem states that all the charges of a charged shell is concentrated at center of shell for points outside the shell and electric field is zero for points inside the charged shell. Therefore, we are shielded from the fields of outside charges when inside a hollow conductor but not shielded when outside a hollow charged conductor.

Q.16 If the surface of a charged conductor is an equipotential, does that mean that charge is distributed uniformly over that surface. If the electric field is constant in magnitude over the surface of a charged conductor, does that mean that charge is distributed uniformly?

Answer: Yes, if the surface of a charged conductor-is an equipotential, it means that charge is distributed uniformly over that surface.

Yes, if the electric field is constant in magnitude over the surface of a charged conductor, it means that charge is distributed uniformly.

Q.17 An electrical worker was accidentally electrocuted, and a newspaper account reported. He accidentally touches a high voltage cable and 20000V of electricity surged through his body. Criticize this statement.

Answer: The statement of newspaper must be as “He accidentally touches a high voltage cable and 20000 V of potential surged through his body”.

Q.18 What is the effect on measured values of potential difference, if we assume that potential of earth is +100 V or -100 V instead of zero.

Answer: The earth is taken as a reference of potential and assigned the value of zero potential. This is taken just for convenience. The other choice would change the potential everywhere by the same amount but would not change the results for the potential difference.

Q.19 Why the electron volt is more convenient unit of energy than the joules?

Answer: The unit eV for energy is a great convenience when dealing with atoms or nuclei in which the charge is easily expressed in terms of e.

Q.20 An electron accelerates from rest through a potential difference V and a proton accelerates from rest through a potential difference -V. Is final speed of electron same as that of proton?

Answer: Since charges on electron and proton have same magnitude, so their kinetic energies are same. As electron is less massive than proton, so speed of electron is greater than that of proton.

Q.21 In one region of space the electric potential has a positive constant value. In another region of space potential has negative constant value. What can be said about the electric field with in each of these two regions of space?

Answer: Electric field intensity is zero in both regions because electric potential has constant value in both regions.

Q.22 Can two different equipotential surfaces intersect?

Answer: Two equipotential surfaces cannot intersect. If they intersect, the point of intersection will have two different values of potential which is not possible.

Q.23 Two like charges at a distance d apart have positive electric potential energy. Conversely, two unlike charges at same distance apart have negative electric potential energy. Give significance of these two statements.

Answer: When two like charges are released, these will move away from on another to infinite separation. When two unlike charges are released, these attract one another. If their separation is to be increased, a positive work must be done.

Q.24 An electron is released from rest in a region of space with non-zero electric field. As electron moves, does it experience a decreasing potential.

Answer: Electric field points in direction of decreasing electric potential and electron moves in a direction opposite to electric field. So, electron experience an increasing potential.

Q.25 If V equals a constant throughout a given region of space, what can you say about E in that region?

Answer: If electric potential  is constant throughout a region, ten electric field intensity  in that region is zero.

Q.26 A spherical balloon contains a positively charged object at its center. As the balloon is inflated to a greater volume while the charged object remains at the center, does the electric potential at the surface of the balloon remain the same? What about electric flux?

Answer: The electric potential is inversely proportional to the radius so it will decrease. Because the same number of field lines passes through a closed surface of any shape or size, the electric flux through the surface remains constant.

Q.27 A negative charge moves in the direction of a uniform electric field. Does the potential energy of system decrease?

Answer: No, the potential energy increases. When a negative charge moves in the direction of the field, i.e. charge moves to a region of lower electric potential. Then the product W=qV increases.

Q.28 Suppose that you follow an electric field line due to a positive point charge. Do electric field intensity and electric potential increases or decreases?

Answer: Electric field intensity and electric potential are given by,

$E=\frac{1}{4\pi\epsilon_\circ}\frac{q}{r^2}$ & $V=\frac{1}{4\pi\epsilon_\circ}\frac{q}{r}$

From above relations, it is clear that if we are moving along path of electric line of force due to a positive charge, both electric field and potential decreases.

Q.29 Two opposite point charges, each of magnitude q are separated by a distance 2r. What is electric potential at a point P midway between them?

Answer: Electric potential at point P due to both charges is,

$$V=\frac{1}{4\pi\epsilon_\circ}\frac{q}{r}+\frac{1}{4\pi\epsilon_\circ}\frac{-q}{r}=0$$

Q.30 Discuss the situation when electric field intensity at a point is zero but electric potential is not zero. Discuss the case when electric potential at a point is zero but electric field intensity is not zero.

Answer: At the midpoint between two equal and like charges, electric field intensity is zero but electric potential is not zero

At midpoint between two equal and opposite charges electric potential is zero but electric field intensity is not zero.

Q.31 Show that $1eV=1.6\times{10}^{-19}J$.

Answer: Kinetic energy of a charges particle in motion is,

Kinertic energy=qV

$$\Rightarrow1eV=1.6\times{10}^{-19}\ C\times1V=1.6\times{10}^{-19}J$$

Q.32 Kinetic energy of a charged particle decreases by 10J as it moves from a point at potential 100V to 200V. What will be the charge on particle?

Answer: Using

$$q\Delta V=E\ \Rightarrow q=\frac{E}{\Delta V}=\frac{10J}{(200-100)V}=0.1C$$

Q.33 If you know E only at a given point, can you calculate V at that point?

Answer: If we know electric field intensity at a appoint, then electric potential cam be calculated by using equation,

$$V_B-V_A=-\int_{A}^{B}{\vec{E}\cdot\vec{dr}}$$

Q.34: A uniform electric field is parallel to the x-axis. In what direction can a charge be moved in this field without any external work being done on the charge?

Answer: The charge can be moved along any path parallel to the yz-plane i.e. perpendicular to the field.

Q.35 Explain why equipotential surfaces are always perpendicular to electric field lines?

Answer: Electric field is related to electric potential by relation,

$$E_x=-\frac{\partial V}{\partial x},\ \ \ \ E_y=-\frac{\partial V}{\partial y},\ \ \ \ \ E_z=-\frac{\partial V}{\partial z}$$

Thus, electric field always point in the direction of the greatest change in electric potential i.e. equipotential surfaces are always perpendicular to electric field lines.

Q.36 Define volt and stat volt. Establish a relation between them.

Answer: The potential at a point is said to be one stat volt if one erg of work is done in taking a positive charge of one stat coulomb from infinity to that point against electric field.

The potential at a point is said to be one volt if one joule of work is done in taking a positive charge of one coulomb from infinity to that point against the electric field. Relation between volt and stat colt is,

$$1V=\frac{1J}{1C}=\frac{{100}^7erg}{3\times{10}^9stat\ coulomb}=\frac{1}{300}\ stat\ volt$$

$$1\ stat\ volt=300\ volts$$

Q.37 In an electron gun, electrons of charge e and mass m are accelerated through a potential difference . Find maximum speed attained by electrons.

Answer: Kinetic energy of electrons is given by,

$$\frac{1}{2}mv^2=eV\ \ \ \ \ \ \ \ \ \Rightarrow v=\sqrt{\frac{2eV}{m}}$$

Q.38 Describe the equipotential surfaces for an infinite line of charge and a uniformly charged sphere.

Answer: The equipotential surfaces are nesting coaxial cylinders around an infinite line of charge. The equipotential surfaces are nesting concentric spheres around a uniformly charged sphere.

Q.39 The potential of a point charge is defined to be zero at an infinite distance. Why can we not define the potential of an infinite line of charge to be zero at r→∞?

Answer: Potential due to infinite line of charge of length is,

$$V=\frac{q}{4\pi\epsilon_\circ L}\ln{\left(\frac{L+\sqrt{r^2+L^2}}{r}\right)}$$

It is clear that potential would diverge to infinity at any finite distance, if it were zero at infinite far away.

Q.40 Given a region in space in which there exist an electrostatic field in z-direction. Show that field is independent of x and y coordinates.

Answer: As electric field is along -axis, so

$$\vec{E}=E\hat{k}$$

Electrostatic field is conservative, so

$$curl\vec{E}=0\ \ \Rightarrow\frac{\partial E}{\partial y}\hat{i}-\frac{\partial E}{\partial x}\hat{j}=0,\ \ \ \ \ \frac{\partial E}{\partial x}=0$$

In other words, electric field does not change with x or y that is independent of x and y coordinates.