Electric Potential Energy and Electric Potential
Electric Potential Energy and Electric Potential
1: Work and Potential Energy Due to Gravity Review
Before discussing energy due to the electrical field, we first review energy due to gravity.
Key components in gravitational potential energy:
Work done by gravitational force on an object when moved.
If an object is moved straight upwards against gravity:
Displacement vector and weight vector are in opposite directions, resulting in negative work.
Change in height is defined as:
\Delta y = yf - yi
Work is calculated as:
W = -w \Delta y = -mg \Delta y (since gravity is conservative, work depends on difference in height).
Change in potential energy derived as:
\Delta PE_g = -W
\Delta PE_g = mg \Delta y (negative from work cancels).
Behavioral implications for potential energy change with height:
If yf > yi (object moves upwards, \Delta y is positive), potential energy increases.
If yf < yi (object moves downwards, \Delta y is negative), potential energy decreases.
If yf = yi (object moves parallel to ground, \Delta y is zero), potential energy remains constant.
General equation for gravitational potential energy:
PEg = mg y or PEg = mg h.
2: Work and Potential Energy For a Charge in a Uniform Electric Field
We can apply the principles of gravitational potential energy to define the electric potential energy of a point charge in a uniform electric field.
Consider a positive charge placed in a uniform electric field created by a capacitor, where the electrical field points in the negative y direction:
The electric force on the positive charge is directed in the negative y direction, similar to the gravitational force discussed previously.
Moving the charge against the electric force leads to negative work:
W = -F_e \Delta y
Where F_e = |q| E, therefore:
W = -|q| E \Delta y (for positive charge moving against the field).
Electric force is also conservative, leading to:
\Delta EPE_e = -W
Thus, \Delta EPE_e = |q| E \Delta y for a positive charge.
Behavioral insights:
If the charge moves upwards against the field, energy increases; if moves downwards with the field, energy decreases; if parallel, energy remains constant.
For a negative charge, the electric force reverses direction, thus results in:
\Delta EPE_e = q E \Delta y.
General equation for displacement against the field:
If moving distance d against the field:
\Delta EPE_e = q E d.
When moving with the field:
\Delta EPE_e = -q E d.
When moving perpendicular to the field:
\Delta EPE_e = 0.
3: General Situation for EPE in Uniform Fields
To analyze uniform fields, we determine how far the charge moves in the direction of the electric field.
Distance with respect to the field is formulated as:
d = r \cos(\theta), where
r is the total distance traveled, and
\theta is the angle relative to the electric field direction.
4: Conceptual Understanding of Electric Potential Energy
The relationship between force and movement direction determines electric potential energy change:
General principle: Potential energy increases when moving opposite to an object's natural direction.
For gravity, moving upwards against gravitational force increases gravitational potential energy.
Electrical analogy:
For a positive charge in an electric field, moving against the field direction (toward another positive charge) increases electric potential energy.
Moving with the field (toward a negative charge) decreases potential energy.
For a negative charge, the opposite is true regarding its potential energy changes.
Conceptual check: If uncertain about energy changes, assess if the charge is moving towards a direction it naturally resists; against the field indicates increasing potential energy.
5: Electric Potential
To define electric potential energy, we need a charge and an electric field, similar to electric force.
Electric potential defined through energy change per unit charge:
\Delta V = \frac{\Delta EPE_e}{q}, applicable universally, not limited to uniform fields.
For uniform electric fields, this yields:
\Delta V = \frac{(q E d)}{q} = E d (against field).
\Delta V = \frac{(-q E d)}{q} = -E d (with field).
\Delta V = 0 (perpendicular to field).
Units of potential:
[V] = 1 \text{ J/C} = 1 \text{ Volt} = 1 V.
Potential as a scalar property of an electric field:
Increases when moving closer to positive charges, and decreases when moving towards negative charges or away from positive charges.
Equipotential lines:
Lines with constant potential, perpendicular to electric field lines, where potential remains unchanged.
Electric potential is a scalar quantity; it has no direction but can be positive, negative, or zero.
6: Equipotential Lines and Field for a Parallel Plate Capacitor
Sketching equipotential lines requires moving parallel to plates, maintaining a constant distance from either plate.
Equipotential lines:
Shown in blue, parallel to plates, increasing in value towards the positive plate, and perpendicular to the electric field.
7: Electric Potential Energy and Conservation of Energy
Electric potential contributes to conservation of energy principles:
Electric force being conservative allows integration with total mechanical energy:
E = PE_g + EPE + PEs + KE, where
KE = \frac{1}{2} m v^2 represents kinetic energy and
potential energies include gravitational, elastic, and electric forms.
Conservation of energy formula remains:
Ef - Ei = \Delta E = W{nc}, where W{nc} pertains to work done by non-conservative forces (e.g., friction).
Additionally, net energy changes summarized as:
\Delta E = \Delta PE_g + \Delta EPE + \Delta PEs + \Delta KE.
8: Relating Change in Potential to Electric Field for Uniform Fields
In uniform fields, moving against the field a distance d results in:
\Delta V = E d.
To determine electric field's magnitude based on potential change during distance d:
E = \frac{|\Delta V|}{d} (while noting that \Delta V is negative when moving with the field).
When traveling opposite to a uniform electric field:
\Delta V = -E d leads to:
E = \frac{|\Delta V|}{d} (ignoring negative sign for magnitude).
Standard equation for uniform electric fields becomes:
E = \frac{|\Delta V|}{d}, with units of
[E] = 1 V/m = 1 N/C.
9: Finding Potential for Point Charges
Point charges create non-uniform electric fields; the field around point charges varies in direction and magnitude based on different positions.
Previous equations apply mainly to uniform fields; hence, we will treat point charges differently through calculus concepts.
Analyzing equipotential lines around a point charge:
If no movement toward or away from a charge occurs, potential remains unchanged.
Equipotential lines form circular patterns around a point charge due to distance parity.
Potential depends on distance (r) from point charge; as you approach a positive charge, the potential rises and must be related inversely to a power of r.
We need to determine the correlations between the charge magnitude and distance.
Dimensional analysis confirms:
V = k \frac{q}{r} fits potential unit requirements:
[V] = \frac{[k][q]}{[r]} = \frac{(N m^2)/C^2 \cdot C/m}{N m/C} allowing for accurate proportionality.
Electric potential equation derived as:
V = k \frac{q}{r}. Potential defined where the measure infinitely far from the charge is zero.
Potential sign interpretations:
Positive charge yields positive V, increasing when closer; negative yields negative V, decreasing closer.
10: Total Electric Potential for Multiple Point Charges
Electric potential as a scalar simplifies calculations for multiple charges.
To find total potential at any point:
Calculate potential from each individual charge and sum:
V = V1 + V2 + V_3 + …
Represented as:
V = k \frac{q1}{r1} + k \frac{q2}{r2} + k \frac{q3}{r3} + …
11: Total Electric Potential for Point Charges on a Circle
Example scenario: Positioning three charges of -q and two of +q on a circle of radius R
Total potential at the center requires summing the contribution:
Each charge is consistent distance R to the center. Thus it follows:
V = k \frac{q}{R} + k \frac{q}{R} - k \frac{q}{R} - k \frac{q}{R} - k \frac{q}{R} results in:
V = -k \frac{q}{R}.
Total electric field calculation would necessitate vector components, showcasing the relative simplicity of potential summation versus field summation.
12: Change In Potential
In uniform fields, moving with the field results in potential change:
\Delta V = -E d.
No straightforward analogy exists for non-uniform fields; thus, when moving from point 1 to point 2 around several point charges, it involves:
Compute potential at point 1 V1 and at point 2 V2 then:
\Delta V = V2 - V1.
13: Potential Energy of a Point Charge in General
From the equation \Delta V = \frac{\Delta EPE}{q}, we derive:
\Delta EPE = q \Delta V.
Potential energy at any point thus expressed as:
EPE = q V, where
V is the potential at that location and
q is an additional charge placed where the potential is defined.
This holds true for both uniform and non-uniform fields.
Example: Given a potential of +1 \times 10^6 V, and placing -2 micro-coulombs results in:
EPE = q V = (-2 \times 10^{-6} C)(+1 \times 10^6 V) = -2 J.
Evaluation of the absolute value of EPE is less significant than changes between points, as emphasized in previous analyses.
Conservation of energy can be applied to examine movement dynamics surrounding charges in complex fields:
E = PE_g + EPE + PEs + KE and
Ef - Ei = \Delta E = W_{nc}.