19.2 Electric Potential in a Uniform Electric Field

Electric Potential in a Uniform Electric Field

Voltage and electric field relationship

A uniform electric field is one where the field has the same magnitude and direction at every point. The most common example is the field between two parallel charged plates. In this kind of field, there's a clean linear relationship between voltage and distance:

$V = Ed$

• $V$ = electric potential difference (voltage) between two points, in volts (V)
• $E$ = electric field strength, in V/m or N/C
• $d$ = distance between the two points measured along the field direction, in meters (m)

Because $E$ is constant everywhere in a uniform field, doubling the distance between two points doubles the voltage between them. This linear relationship is what makes uniform-field problems straightforward compared to fields that vary with position.

Where the equation comes from

The relationship $V = Ed$ isn't just handed to you; it comes from the work-energy connection.

1. When a charge $q$ moves through a uniform electric field, the field does work on it: $W = qEd$.

2. That work equals the change in the charge's electric potential energy: $W = q(V_i - V_f)$.

3. Setting these equal gives $qEd = q(V_i - V_f)$.

4. The $q$ cancels, leaving $Ed = V_i - V_f$, which is the potential difference $V$ between the two points.

So $V = Ed$ is really a statement about how much work the field does per unit charge over a given distance.

Energy and work in electric fields

• The work done by the electric field on a charge equals the negative change in that charge's potential energy: $W = -\Delta PE$. If the field does positive work, the charge loses potential energy (and gains kinetic energy).
• Conservation of energy still holds. A charge speeding up in an electric field is converting potential energy into kinetic energy, just like a ball rolling downhill converts gravitational PE into KE.
• Equipotential surfaces are surfaces where every point has the same electric potential. In a uniform field, these are flat planes parallel to the plates, and they're always perpendicular to the field lines.
• An electric dipole placed in a uniform field experiences a torque that tries to align it with the field. The dipole has lowest potential energy when aligned and highest when pointing opposite to the field.

Calculations in uniform fields

You can rearrange $V = Ed$ to solve for whichever variable is unknown:

1. Find the field strength given voltage and distance: $E = \frac{V}{d}$
2. Find the voltage given field strength and distance: $V = Ed$
3. Find the distance given voltage and field strength: $d = \frac{V}{E}$

A few things to watch for when solving problems:

• The units V/m and N/C are equivalent for electric field strength. You'll see both, and they mean the same thing.
• The distance $d$ must be measured along the direction of the electric field, not at an angle to it. If a charge moves diagonally between the plates, you only use the component of displacement parallel to $E$.
• For a parallel plate capacitor with plate separation 0.02 m and a voltage of 100 V across the plates, the field strength is $E = \frac{100}{0.02} = 5000 \text{ V/m}$. That's a typical scale for these problems.