Hello Future Physicist! Introduction to Electric Potential
Welcome to the chapter on Electric Potential! This is where we shift our focus in electricity from considering forces (like Coulomb’s Law and Electric Field Strength, \(E\)) to considering energy.
Don't worry if this feels a bit abstract at first—we’re just using the concept of energy we learned in Mechanics and applying it to electric charges. Understanding potential is crucial, as it forms the bedrock for studying capacitance and circuits later on!
Key Takeaway from this Section:
- Electric Field Strength (\(E\)) tells us about the force a charge experiences (a vector quantity).
- Electric Potential (\(V\)) tells us about the energy a charge possesses (a scalar quantity).
1. Defining Electric Potential (\(V\))
In simple terms, electric potential is a way to describe how much energy an electric field stores at a specific point.
What is Absolute Electric Potential?
The absolute electric potential (\(V\)) at a point is formally defined as the work done per unit positive test charge in moving that charge from infinity to that point.
Key Concept: Zero Potential at Infinity
To define potential, we need a reference point. Just like when measuring gravitational potential energy, we choose ground level as \(GPE = 0\), for electric potential, we choose a point far, far away from the charge—called infinity (\(r = \infty\)).
- At infinity, the electric field from the source charge is zero, so the potential (\(V\)) is also defined as zero.
The Analogy: Electric Potential vs. Gravitational Potential
If you find electric potential tricky, think about gravity:
- Lifting a heavy object (like a bowling ball) against gravity takes energy (work done). The ball gains Gravitational Potential Energy.
- Moving a positive charge towards a positive source charge takes energy (work done) against the electrostatic force. The charge gains Electric Potential.
The unit for electric potential is the Volt (\(V\)), which is equivalent to one Joule per Coulomb (\(J C^{-1}\)).
Quick Review: Key Terms
Electric Potential (\(V\)) is a scalar quantity (it only has magnitude, not direction).
2. Electric Potential Difference (\(\Delta V\))
More often in physics problems, we deal with the change in potential between two measurable points, known as Electric Potential Difference (PD), or simply voltage.
Work Done and Potential Difference
The PD between two points (A and B) is the work done (\(\Delta W\)) moving a unit positive charge (\(Q\)) from point A to point B.
We can rearrange this definition to calculate the work done in moving a charge:
$$ \Delta W = Q \Delta V $$
Where:
- \(\Delta W\) is the Work Done (Energy transferred), measured in Joules (J).
- \(Q\) is the Charge being moved, measured in Coulombs (C).
- \(\Delta V\) is the Potential Difference between the two points, measured in Volts (V).
Imagine pushing a small cart (charge \(Q\)) up a ramp. The height you lift it is the potential difference (\(\Delta V\)), and the total energy you expend is the work done (\(\Delta W\)).
Common Pitfall: Sign Confusion!
Always consider the sign of the charge \(Q\) and whether the potential is increasing or decreasing:
- Moving a positive charge: Moving it against the field (e.g., towards a positive plate) requires external work, so \(\Delta W\) is positive.
- Moving an electron (negative charge): If you move an electron (negative \(Q\)) into a region of higher potential, the field does positive work on it, and its potential energy decreases. Be careful with the signs when calculating \(\Delta W = Q\Delta V\)!
3. Electric Potential in a Radial Field (Point Charges)
When dealing with a single point charge (or a uniformly charged sphere, where the charge is treated as being at the centre), the field is radial (pointing outwards or inwards). We have a specific formula for calculating the potential \(V\) at a distance \(r\) from the source charge \(Q\).
The Radial Potential Equation
The magnitude of the Electric Potential (\(V\)) at a distance \(r\) from a point charge \(Q\) is given by:
$$ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} $$
Where:
- \(\epsilon_0\) is the permittivity of free space (a constant).
- \(Q\) is the source charge (Crucially, \(Q\) includes its sign—positive or negative).
- \(r\) is the distance from the charge.
Did You Know? Gravity vs. Electric Fields
Comparing the gravitational potential \(V_g = -GM/r\) and electric potential \(V = kQ/r\), notice a crucial difference: Gravitational forces are only attractive, so \(V_g\) is always negative (by convention). Electric forces can be attractive or repulsive, so \(V\) can be positive (for a positive source charge, \(+Q\)) or negative (for a negative source charge, \(-Q\)).
Variation of Potential with Distance (\(V\) vs. \(r\))
Because \(V\) is proportional to \(1/r\), the potential drops off slower than the electric field strength \(E\), which is proportional to \(1/r^2\).
- For a positive source charge, \(V\) is always positive. The graph of \(V\) against \(r\) starts high near the charge and decreases towards zero as \(r \to \infty\).
- For a negative source charge, \(V\) is always negative. The graph starts low (very negative) near the charge and increases towards zero as \(r \to \infty\).
4. Equipotential Surfaces
Just as topographical maps show lines of constant height, electric fields have surfaces that show constant potential. These are called equipotential surfaces.
Definition and Properties
An equipotential surface is a surface (or line, in 2D diagrams) connecting all points in an electric field that have the same electric potential.
Key Property: No Work Done
When a charge moves along an equipotential surface, the potential difference (\(\Delta V\)) between the start and end point is zero. Therefore, based on \(\Delta W = Q\Delta V\):
$$ \Delta W = 0 $$
No net work is done by or against the field when moving a charge along an equipotential surface.
Relationship to Field Lines
Equipotential surfaces are always perpendicular (at 90°) to the electric field lines (\(E\)).
- Analogy: If electric field lines are like lines of steepest slope down a hill, equipotential surfaces are like contour lines running flat around the hill—you walk along a contour line without going up or down.
- Where field lines are closely spaced (strong \(E\)), the equipotential surfaces are also closely spaced.
5. The Link Between \(E\) and \(V\): The Potential Gradient
Electric potential (\(V\)) and electric field strength (\(E\)) are intrinsically linked. \(E\) is essentially a measure of how quickly \(V\) changes over distance.
E is the Potential Gradient
The Electric Field Strength (\(E\)) is defined as the negative potential gradient.
$$ E = - \frac{\Delta V}{\Delta r} $$
Where:
- \(\Delta V\) is the change in potential.
- \(\Delta r\) is the distance moved in the direction of the field.
The negative sign is vital and simply means that the electric field (\(E\)) points in the direction of decreasing potential.
The unit relationship shows this clearly: \(E\) is measured in \(\text{N C}^{-1}\), which is dimensionally equivalent to the potential gradient unit: \(\text{V m}^{-1}\).
Using Graphs to Relate E and V
1. Finding \(E\) from a \(V\)-r graph
Since \(E = - \frac{\Delta V}{\Delta r}\):
- The magnitude of \(E\) is the magnitude of the gradient of the \(V\) against \(r\) graph.
2. Finding \(\Delta V\) from an \(E\)-r graph
If we rearrange the gradient equation: \(\Delta V = -E \Delta r\). If we sum up many small \(E \Delta r\) segments:
- The change in potential (\(\Delta V\)) between two points is found by calculating the area under the Electric Field Strength (\(E\)) against distance (\(r\)) graph.
Memory Aid: Think about the units: If you multiply the units of \(E\) (\(V/m\)) by the units of \(r\) (\(m\)), you get the units of \(V\) (\(V\)). So, Area under E-r graph must equal potential difference!
Chapter Summary: Key Takeaways
- Electric Potential \(V\): Work done per unit positive charge moving from infinity to a point. It is a scalar quantity.
- Zero Reference: \(V = 0\) is defined at infinity.
- Work Done: The energy needed to move a charge \(Q\) between points separated by potential difference \(\Delta V\) is given by \(\Delta W = Q\Delta V\).
- Radial Field Potential: \(V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}\). Note that \(V\) is proportional to \(1/r\).
- Equipotentials: Surfaces of constant potential. Moving a charge along them requires no work. They are always perpendicular to electric field lines.
- Potential Gradient: The Electric Field Strength is the negative gradient of the potential: \(E = - \frac{\Delta V}{\Delta r}\).
- Graphical Link: The area under an \(E\)-r graph gives the potential difference (\(\Delta V\)).
You’ve conquered the concept of potential! Moving from forces to energy is a massive step in mastering electrostatics. Keep practicing those calculations!