Magnetic Field from Electric Field Calculator

Determine the induced magnetic field based on a changing electric field and distance.

Induced Magnetic Field (B)

0 T

Calculation Details:

What is Calculating a Magnetic Field from an Electric Field?

One of the most profound discoveries in physics is that electricity and magnetism are two sides of the same coin: the electromagnetic force. While stationary electric charges create electric fields and moving charges (currents) create magnetic fields, the connection goes even deeper. A changing electric field in a region of space can itself induce a magnetic field, even in the complete absence of moving charges. This concept, proposed by James Clerk Maxwell, was the missing piece that unified electromagnetism and predicted the existence of electromagnetic waves, like light.

To calculate the magnetic field using an electric field, you must focus on the rate of change of the electric field (often denoted as dE/dt). A static, unchanging electric field will not produce a magnetic field. However, if the electric field’s strength is increasing or decreasing, it creates a “displacement current” which in turn generates a circulating magnetic field around it. This principle is fundamental to how capacitors work in AC circuits and how radio waves propagate through space. For more on fundamental forces, see our article on the four fundamental forces.

Formula to Calculate Magnetic Field from Electric Field

The relationship is described by the Ampere-Maxwell equation. For a common symmetrical case, like the field inside a charging circular capacitor, we can derive a more direct formula. The induced magnetic field (B) at a distance (r) from the center is given by:

B = (μ₀ * ε₀ * r / 2) * (dE/dt)

This formula shows that the magnetic field’s strength is directly proportional to both the rate of change of the electric field and the distance from the center.

Variables in the Formula

Variable	Meaning	SI Unit	Typical Range
B	Induced Magnetic Field	Tesla (T)	10⁻⁹ T to 10⁻³ T
μ₀	Permeability of Free Space	Henry per meter (H/m)	~1.257 x 10⁻⁶ H/m (Constant)
ε₀	Permittivity of Free Space	Farad per meter (F/m)	~8.854 x 10⁻¹² F/m (Constant)
r	Radial Distance	meters (m)	10⁻³ m to 1 m
dE/dt	Rate of Electric Field Change	Volts per meter-second (V/m·s)	10¹⁰ V/m·s to 10¹⁵ V/m·s

For more details on the constants involved, you can explore our guide on physical constants.

Practical Examples

Example 1: Inside a Capacitor

Imagine you are analyzing a parallel-plate capacitor being charged. The electric field between the plates is changing rapidly.

• Inputs:
  • Rate of Electric Field Change (dE/dt): 2.0 x 10¹³ V/m·s
  • Distance from Center (r): 0.05 meters (5 cm)
• Calculation:
  • B = ( (1.257e-6) * (8.854e-12) * 0.05 / 2 ) * (2.0e13)
  • B ≈ 5.56 x 10⁻⁶ T or 5.56 microteslas (µT)
• Result: The induced magnetic field at 5 cm from the center is approximately 5.56 µT.

Example 2: A High-Frequency Scenario

Consider a scenario with a very rapidly oscillating electric field, such as one might find in a high-frequency antenna system.

• Inputs:
  • Rate of Electric Field Change (dE/dt): 8.0 x 10¹⁴ V/m·s
  • Distance from Center (r): 0.01 meters (1 cm)
• Calculation:
  • B = ( (1.257e-6) * (8.854e-12) * 0.01 / 2 ) * (8.0e14)
  • B ≈ 4.45 x 10⁻⁵ T or 44.5 microteslas (µT)
• Result: The induced magnetic field at 1 cm from the center is approximately 44.5 µT. Learn about how this relates to wave propagation in our article about understanding electromagnetic waves.

How to Use This Magnetic Field Calculator

This tool makes it simple to calculate magnetic field using electric field data. Follow these steps:

1. Enter dE/dt: In the first field, input the rate of change of the electric field in Volts per meter per second.
2. Enter Distance: In the second field, input the radial distance from the center of the field.
3. Select Units: Use the dropdown to select the unit for your distance (meters, centimeters, or millimeters). The calculation will automatically convert it to meters.
4. Interpret the Results: The calculator instantly displays the induced magnetic field (B) in Tesla (T). It also shows the formula used with your specific inputs.
5. Analyze the Chart: The dynamic chart visualizes how the magnetic field strength changes with distance for your given dE/dt, illustrating their linear relationship.

Key Factors That Affect Induced Magnetic Fields

Several factors influence the strength of the induced magnetic field:

• Rate of Electric Field Change (dE/dt): This is the most critical factor. A faster change in the E-field induces a stronger B-field. A static E-field (dE/dt = 0) induces no magnetic field.
• Distance from the Center (r): The induced magnetic field is zero at the center (r=0) and increases linearly as you move outward.
• Medium’s Permittivity (ε): The calculator uses the vacuum permittivity (ε₀). If the field is in a dielectric material, its higher permittivity would slightly increase the resulting B-field. Our dielectric constant calculator can provide more insight.
• Medium’s Permeability (μ): Similarly, the calculator uses vacuum permeability (μ₀). A magnetic material would dramatically alter the outcome.
• Symmetry: The formula used here assumes cylindrical symmetry, such as in the center of a parallel-plate capacitor. For less symmetric fields, the calculation becomes a more complex vector problem.
• Presence of Conduction Currents: This calculator only accounts for the “displacement current” from a changing E-field. A complete Ampere-Maxwell law calculation would also include any real currents from moving charges.

Frequently Asked Questions (FAQ)

1. What is displacement current?

Displacement current is the term Maxwell used for the effect of a time-varying electric field. It’s not a real current of moving charges, but it produces a magnetic field just as a real current does.

2. Does a static electric field create a magnetic field?

No. Only a changing electric field (dE/dt ≠ 0) can induce a magnetic field. This is a fundamental principle of electromagnetism.

3. Why is the magnetic field zero at the center (r=0)?

In a cylindrically symmetric system, the magnetic field lines form closed loops around the central axis. At the exact center, the “loop” has zero radius, so the field strength is zero.

4. What unit is the result in?

The result for the magnetic field (B) is given in Tesla (T), the standard SI unit for magnetic flux density.

5. Can I use this for any shape of electric field?

No. This specific formula is an accurate simplification for systems with high cylindrical symmetry, like the area between the plates of a large circular capacitor. For other geometries, the relationship is more complex.

6. How does this relate to light?

Light is an electromagnetic wave. It consists of a changing electric field and a changing magnetic field that continually induce each other as they travel through space. This calculator models one half of that process. Check out our refractive index calculator to learn more about light.

7. What are μ₀ and ε₀?

μ₀ (mu-nought) is the magnetic permeability of free space, and ε₀ (epsilon-nought) is the electric permittivity of free space. They are fundamental physical constants that describe how fields propagate in a vacuum.

8. Is the induced magnetic field constant?

No, if the rate of change of the electric field (dE/dt) is itself changing, the induced magnetic field will also change in response, which would then induce a new electric field, and so on—this is the basis of an electromagnetic wave.

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