Understanding Electric Field Lines from a Charged Conducting Sphere
Let's analyze the properties of electric field lines, especially in the context of a positively charged conducting sphere, to determine their behavior.
Electric field lines are a visual representation of the electric field in a region. They indicate the direction of the force that would be exerted on a small positive test charge placed at that point. Several important rules govern electric field lines:
- Electric field lines originate from positive charges and terminate on negative charges or extend to infinity.
- The density of electric field lines represents the strength of the electric field; closer lines indicate a stronger field.
- Electric field lines never cross each other.
- Inside a conductor in electrostatic equilibrium, the net electric field is always zero.
- The surface of a conductor in electrostatic equilibrium is an equipotential surface.
- Electric field lines must be perpendicular to equipotential surfaces. Therefore, electric field lines are always perpendicular (at right angles) to the surface of a conductor in electrostatic equilibrium.
Analyzing the Positively Charged Conducting Sphere
Consider an isolated positively charged conducting sphere in electrostatic equilibrium. The positive charges reside on the outer surface of the sphere due to mutual repulsion and the fact that the field is zero inside.
Based on the rules above:
- Since it is a conducting sphere in electrostatic equilibrium, the electric field lines must be perpendicular to its surface. This eliminates options that suggest lines are tangential or at any angle other than 90 degrees.
- Since the sphere is positively charged, the electric field lines must originate from its surface and point outwards, away from the sphere.
- For an isolated sphere, due to spherical symmetry, the electric field outside the sphere is radial, meaning it points directly outwards from the center of the sphere (as if all the charge were concentrated at the center).
Combining these points, the electric field lines leave the surface of the positively charged conducting sphere perpendicularly and point outwards, following radial paths away from the center.
Evaluating the Given Options
Let's examine each option in light of these principles:
- Tangential to the conducting surface: Incorrect. Field lines must be perpendicular to the surface of a conductor in equilibrium.
- At right angles to the conducting surface and towards the centre of the sphere: Incorrect. For a positively charged sphere, the field lines point outwards, away from the charge, not towards the center.
- At any angle to the conducting surface: Incorrect. Field lines must be at right angles (perpendicular) to the surface of a conductor in equilibrium.
- At right angles to the conducting surface and outwards from the centre of the sphere: Correct. This option correctly describes both conditions: perpendicularity to the conductor's surface and the outward direction characteristic of a positive charge, appearing radial from the center due to spherical symmetry.
Therefore, the electric field lines from an isolated positively charged conducting sphere are at right angles to the conducting surface and outwards from the center of the sphere.
Revision Table: Electric Field Properties
| Property | Electric Field Lines Behavior |
|---|---|
| Origin/Termination | Originate from positive charges, terminate on negative charges or infinity. |
| Conductors in Equilibrium | Field is zero inside; lines are perpendicular to the surface. |
| Positive Charge | Lines point outwards away from the charge. |
| Negative Charge | Lines point inwards towards the charge. |
| Equipotential Surfaces | Lines are always perpendicular to equipotential surfaces. |
Additional Information: Electrostatic Equilibrium and Conductors
When a conductor is in electrostatic equilibrium (meaning charges are not moving), two key conditions hold true:
- The electric field inside the conductor is zero. Any net electric field inside would cause charges to move, contradicting the state of equilibrium.
- Any net charge on a conductor resides entirely on its surface. This is a consequence of Gauss's Law and the fact that the field is zero inside.
The electric field just outside the surface of a charged conductor is given by $\vec{E} = \frac{\sigma}{\epsilon_0} \hat{n}$, where $\sigma$ is the surface charge density and $\hat{n}$ is a unit vector normal to the surface. This formula explicitly shows that the electric field is perpendicular (normal) to the surface.
Important Questions from Applications of Gauss’s Law
A charge Q is placed at the centre of a cube. Find the flux of the electric field through the six surfaces of the cube.
- An infinitely long straight uniformly charged wire has a linear charge density of $\lambda$. Calculate the work done by the electric field when a point charge $q$ is moved from an initial distance $r_1$ to a final distance $r_2$ ($r_2 > r_1$) from the wire.
The variation of electric field with respect to distance from centre of a charged conducting spherical shell of radius R is given by :
An infinitly long wire is charged uniformly with charge density λ and placed in air, the electric field at distance r from wire will be:
According to Gauss’s law, the electric field due to an infinitely long thin charged wire varies as:
