The relationship between electric field lines and equipotential surfaces is a fundamental concept in electrostatics. It can be understood by considering the definition of an equipotential surface and the work done by the electric field.
Definition of Equipotential Surface: An equipotential surface is a surface over which the electric potential is constant. This means that for any two points A and B on such a surface, the potential difference \(V_B - V_A = 0\).
Work Done by Electric Field: The work done by an electric field \(\vec{E}\) in moving a charge \(q\) from point A to point B is given by the negative of the change in potential energy, or \(W_{AB} = -q(V_B - V_A)\). Alternatively, it can be expressed as the line integral of the electric force: \(W_{AB} = \int_A^B \vec{F} \cdot d\vec{l} = \int_A^B q\vec{E} \cdot d\vec{l}\).
Work Done on an Equipotential Surface: If a charge is moved along an equipotential surface, the potential difference between any two points on that surface is zero (\(V_B - V_A = 0\)). Therefore, the work done by the electric field in moving a charge along an equipotential surface must also be zero:
\[ W_{AB} = q(V_A - V_B) = q(0) = 0 \]
Relationship between Electric Field and Displacement: Since \(W_{AB} = \int_A^B q\vec{E} \cdot d\vec{l} = 0\), and \(q\) is not zero, it implies that \(\int_A^B \vec{E} \cdot d\vec{l} = 0\). For this to be true for any arbitrary path segment \(d\vec{l}\) on the equipotential surface, the electric field \(\vec{E}\) must be perpendicular to the displacement vector \(d\vec{l}\) at every point on the surface. This is because the dot product \(\vec{E} \cdot d\vec{l} = |\vec{E}| |d\vec{l}| \cos\theta\), and for it to be zero when neither \(\vec{E}\) nor \(d\vec{l}\) are zero, \(\cos\theta\) must be zero, which means \(\theta = 90^\circ\).
Conclusion: Electric field lines represent the direction of the electric field. Since the electric field vector \(\vec{E}\) is perpendicular to any displacement vector \(d\vec{l}\) on an equipotential surface, the electric field lines must be perpendicular to the equipotential surfaces.
Correct Option: B) They are perpendicular
This is the correct relationship derived from the definition of equipotential surfaces and the
The question asks about the work done in moving a charge along an equipotential surface. We need to recall the definition of an equipotential surface and the formula for work done in an electric field.
C) It is zero. As derived above, since the potential difference between any two points on an equipotential surface is zero, the work done in moving a charge between those points is also zero.
Equipotential surfaces are surfaces where the electric potential is constant. The electric field lines are always perpendicular to these surfaces. The direction of movement of equipotential surfaces relative to a source charge depends on whether the potential is increasing or decreasing, which in turn depends on the sign of the source charge and the direction of the electric field.
D) Both toward and away from the charge
Equipotential surfaces exist at various distances from the source charge, corresponding to different potential values. Depending on whether you are considering a higher or lower potential value relative to a reference, the equipotential surface will be either closer to (toward) or farther from (away from) the source charge. For instance, if you move from a high potential to a low potential, you might move away from a positive charge or toward a negative charge. Conversely, moving from a low potential to a high potential would involve moving toward a positive charge or away from a negative charge. Therefore, equipotential surfaces are found in both directions relative to the charge.
The question asks about the physical significance of the spacing between equipotential lines. Equipotential lines are lines connecting points of equal electric potential. The relationship between electric potential and electric field strength is key to understanding this.
Correct Option: B) The strength of the electric field
The question asks about the electric field inside a conductor that is in electrostatic equilibrium. Electrostatic equilibrium means that there is no net movement of charge within the conductor. We need to recall the fundamental properties of conductors in such a state.
Therefore, for a conductor in electrostatic equilibrium, the electric field inside it must be zero.
B) It is zero — As explained above, if there were any electric field inside a conductor in electrostatic equilibrium, free charges would move, creating a current. This contradicts the definition of electrostatic equilibrium. Thus, the electric field inside must be zero.
This question pertains to the fundamental properties of conductors in electrostatic equilibrium. We need to recall the behavior of charges within a conductor when no current is flowing.
Correct Option: B) On the surface of the conductor. In electrostatic equilibrium, any excess charge on a conductor resides entirely on its outer surface. This is because the free charges repel each other and distribute themselves to minimize their potential energy, which occurs when they are as far apart as possible, i.e., on the surface. Also, the electric field inside a conductor in electrostatic equilibrium is zero, which implies no net charge can exist within its volume.