Formula Used
The polarization vector is uniform. The potential is measured from zero at infinity. The angle θ is measured from the polarization axis.
Inside the sphere, r < a:
V = P r cos(θ) / (3 ε)
Outside the sphere, r ≥ a:
V = P a³ cos(θ) / (3 ε r²)
Here, ε = ε0 εr. The equivalent dipole moment is p = 4πa³P / 3.
How to Use This Calculator
Enter the uniform polarization in C/m². Add the sphere radius and point distance. Choose matching length units. Enter the polar angle in degrees.
Select a medium preset or choose custom relative permittivity. Press the submit button. The result appears above the form and below the header. Use the CSV or PDF buttons after a valid calculation.
Example Data Table
| Case |
P C/m² |
a m |
r m |
θ deg |
εr |
Region |
Potential V |
| 1 |
0.002 |
0.05 |
0.02 |
0 |
1 |
Inside sphere |
1.505879E+6 |
| 2 |
0.002 |
0.05 |
0.08 |
35 |
1 |
Outside sphere |
1.204632E+6 |
| 3 |
0.004 |
0.1 |
0.1 |
60 |
1.0006 |
Outside sphere |
7.524879E+6 |
| 4 |
0.001 |
0.03 |
0.01 |
90 |
5 |
Inside sphere |
4.610424E-12 |
| 5 |
0.006 |
0.06 |
0.15 |
120 |
80 |
Outside sphere |
-1.355291E+4 |
Understanding the Physical Model
A uniformly polarized sphere is a classic electrostatics problem. The polarization vector is constant everywhere inside the sphere. It points along one chosen axis. This calculator uses the polar angle measured from that axis. The result is the scalar electric potential at the selected point.
Why the Potential Changes by Region
The sphere has bound charge, not free charge. The volume bound charge is zero when polarization is uniform. The surface bound charge varies with angle. It is strongest near the poles and zero near the equator. Outside the sphere, the field acts like a centered electric dipole. Inside the sphere, the potential changes linearly with the position component along the polarization direction.
Reference and Sign Meaning
The usual reference sets potential to zero at infinity. With that choice, the outside expression naturally falls as distance increases. Positive potential appears near the positive polarization side. Negative potential appears on the opposite side. At the equator, the cosine term is zero, so the potential is zero for both regions.
Using the Result Carefully
The formulas assume a perfect sphere and uniform polarization. They also assume a static field. Edge defects, nearby conductors, and external fields are ignored. The selected permittivity controls the scale of the answer. A larger relative permittivity gives a smaller potential. This is useful when comparing vacuum, air, and other simple media.
Engineering and Study Uses
Students can use the tool to check textbook problems. Instructors can create sample data quickly. Designers can estimate ideal dipole behavior before using simulation software. The dipole moment output also helps compare the polarized sphere with compact source models.
Practical Interpretation
Use radius and distance in consistent units. The calculator converts common length units to meters. Enter polarization in coulombs per square meter. Choose the point angle with respect to the polarization axis. When the point distance is smaller than the radius, the inside formula is used. When it equals or exceeds the radius, the outside formula is used. The boundary remains continuous, which is an important check on the calculation.
Limit Notes
Results are estimates. They are best for learning, comparison, and first checks. For complex materials, confirm the answer with boundary-value methods and numerical simulation.
FAQs
1. What does uniformly polarized mean?
It means the polarization vector has the same magnitude and direction at every point inside the sphere. This removes volume bound charge and leaves only angle-dependent surface bound charge.
2. Which angle should I enter?
Enter the polar angle measured from the direction of polarization. A point on the positive polarization axis has θ = 0°. A point on the equator has θ = 90°.
3. Why are there two formulas?
The field behavior changes at the sphere boundary. Inside, potential changes linearly with r cos(θ). Outside, the sphere behaves like an ideal electric dipole centered at the origin.
4. What happens at the sphere surface?
At r = a, the inside and outside potential expressions give the same value. This continuity is expected for the electric potential across the polarized surface.
5. Can polarization be negative?
Yes. A negative value reverses the sign of the potential for the same point and angle. It represents polarization along the opposite chosen axis.
6. Why does permittivity matter?
Permittivity controls how strongly electric potential develops for a given bound charge distribution. Higher relative permittivity reduces the calculated potential in this ideal model.
7. Is the outside result a dipole result?
Yes. For points outside the sphere, the potential is equivalent to the potential of a centered dipole with moment p = 4πa³P / 3.
8. Can I use this for real materials?
Use it as an ideal estimate. Real samples may have nonuniform polarization, nearby boundaries, anisotropy, defects, or external fields. Those cases may need numerical modeling.