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:

    • $\Delta y = y<em>f - y</em>i$

  • 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 $y<em>f = y</em>i$ (object moves parallel to ground, $\Delta y$ is zero), potential energy remains constant.

• General equation for gravitational potential energy:

  • $PE<em>g = mg y$ or $PE</em>g = mg h$.

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:

  • $E<em>f - E</em>i = \Delta E = W<em>{nc}$, where $W</em>{nc}$ 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:

  • $V = V<em>1 + V</em>2 + V_3 + …$

  • Represented as:

  • $V = k \frac{q<em>1}{r</em>1} + k \frac{q<em>2}{r</em>2} + k \frac{q<em>3}{r</em>3} + …$

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 $V<em>1$ and at point 2 $V</em>2$ then:

  • $\Delta V = V<em>2 - V</em>1$.

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

  • $E<em>f - E</em>i = \Delta E = W_{nc}$.