How To Find Magnetic Field From Electric Field:With Problems

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Understanding the relationship between electric fields and magnetic fields is crucial in the study of electromagnetism. In this blog post, we will delve into the topic of how to find the magnetic field from an electric field. We will explore the concepts, mathematical representations, and examples to help you grasp this fundamental connection between these two fields.

 

Magnetic Field From Electric field

How to Calculate the Electric Field

A. The Concept of Electric Field

Before we dive into finding the magnetic field from an electric field, let’s start by understanding the concept of the electric field itself. electric field refers to the region surrounding an electric charge where another charge experiences a force. It is a fundamental concept in electromagnetism and helps us understand the interaction between charges.

B. The Mathematical Representation of Electric Field

To calculate the electric field at a given point in space, we use the formula:

E = \frac{kQ}{r^2}

In this equation, (E) represents the electric field, (k) is the electrostatic constant k \approx 9 \times 10^9\, Nm^2/C^2, (Q) is the charge creating the electric field, and (r) is the distance from the charge to the point where we want to calculate the electric field.

C. Worked Out Examples on Calculating Electric Field

Magnetic Field From Electric field

Let’s work through a couple of examples to illustrate the process of calculating the electric field.

Example 1: Suppose we have a positive charge (Q) of 2 μC located 4 meters away from a test charge. What is the electric field at that point?

Using the formula, we can substitute the values into the equation:

E = \frac{(9 \times 10^9\, Nm^2/C^2) \times (2 \times 10^{-6}\, C)}{(4\, m)^2}

Simplifying the equation, we find that the electric field is 112.5 N/C.

Example 2: Let’s consider a different scenario where we have two charges, +3 μC and -5 μC, located 2 meters and 5 meters away from a point, respectively. What is the net electric field at that point?

To calculate the net electric field, we need to consider the contributions from both charges. We use the principle of superposition, which states that the net electric field is the vector sum of the individual electric fields.

 

E_{\text{net}} = E_1 + E_2

Substituting the values into the equation:

E_{\text{net}} = \frac{(9 \times 10^9\, Nm^2/C^2) \times (3 \times 10^{-6}\, C)}{(2\, m)^2} + \frac{(9 \times 10^9\, Nm^2/C^2) \times (-5 \times 10^{-6}\, C)}{(5\, m)^2}

After evaluating the expression, we find that the net electric field is approximately -10.8 N/C.

How to Find the Magnetic Field from Electric Field

how to find magnetic field from electric field
Image by Openstax – Wikimedia Commons, Licensed under CC BY-SA 4.0.
how to find magnetic field from electric field
Image by Geek3 – Wikimedia Commons, Licensed under CC BY-SA 3.0.

A. Understanding the Connection between Electric Field and Magnetic Field

Now that we have a good grasp of the electric field, let’s explore the connection between the electric field and the magnetic field. According to Maxwell’s Equations, a changing electric field induces a magnetic field. This phenomenon is known as electromagnetic induction and is the basis for various technologies, such as electric generators.

B. The Mathematical Approach to Finding Magnetic Field from Electric Field

To find the magnetic field from an electric field, we can use Faraday’s Law of Electromagnetic Induction. The law states that the induced electromotive force (EMF) around a closed loop is equal to the negative rate of change of the magnetic flux through the loop.

Mathematically, Faraday’s Law can be expressed as:

\nabla \times E = -\frac{\partial B}{\partial t}

In this equation, (E) represents the electric field, (B) is the magnetic field, and (\nabla \times E) denotes the curl of the electric field.

C. Worked Out Examples on Finding Magnetic Field from Electric Field

Let’s work through a couple of examples to illustrate how to find the magnetic field from an electric field.

Example 1: Consider a situation where an electric field is changing at a rate of 5 V/m/s. What is the magnitude of the induced magnetic field?

Using Faraday’s Law, we can set the rate of change of the magnetic field frac{\partial B}{\partial t} equal to the given value of 5 V/m/s:

\frac{\partial B}{\partial t} = -5\, V/m/s

Example 2: Let’s explore another scenario. Suppose the electric field is given by (E = 2x\, V/m), where (x) is the distance from the origin along the x-axis. What is the magnetic field at a point (2, 0, 0)?

To find the magnetic field, we need to calculate the curl of the electric field. In this case, since (E) only has an x-component, the curl simplifies to:

(\nabla \times E = \begin{vmatrix} i & j & k \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ 2x & 0 & 0 \end{vmatrix} = 0\, k \, V/m)

Thus, the magnetic field at the given point is zero.

How to Determine the Direction of Magnetic Field from Electric Field

how to find magnetic field from electric field
Image by File:VFPt dipoles magnetic.svg – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.

A. The Concept of Direction in Magnetic Field

Magnetic Field From Electric field

Determining the direction of the magnetic field resulting from an electric field is crucial to fully understand the electromagnetic phenomenon. The direction of the magnetic field is perpendicular to both the electric field and the direction of the changing electric field.

 

B. The Method to Determine the Direction of Magnetic Field from Electric Field

To determine the direction of the magnetic field, we can use the right-hand rule. If you extend your right hand with the fingers pointing in the direction of the electric field and curl your fingers towards the changing electric field, your thumb will point in the direction of the induced magnetic field.

C. Worked Out Examples on Finding the Direction of Magnetic Field from Electric Field

Let’s work through an example to illustrate how to determine the direction of the magnetic field from an electric field.

Example: Consider a scenario where an electric field is directed along the x-axis, and the changing electric field is in the positive y-direction. What is the direction of the induced magnetic field?

Using the right-hand rule, if we point our fingers in the direction of the electric field (along the x-axis) and then curl our fingers towards the positive y-direction (the changing electric field), our thumb will point in the positive z-direction. Therefore, the induced magnetic field will be directed along the positive z-axis.

VI. How a Changing Electric Field Produces a Magnetic Field

A. Understanding the Phenomenon of Induction

Now let’s explore the phenomenon of induction in greater detail. Induction occurs when a changing electric field generates a magnetic field. This process is responsible for the operation of various devices, such as transformers and electric motors.

B. The Role of Changing Electric Field in Producing Magnetic Field

When the electric field changes over time, it creates a circulating magnetic field around it. The magnitude and direction of the magnetic field depend on the rate and nature of the change in the electric field.

C. Worked Out Examples on How Changing Electric Field Produces Magnetic Field

To gain a better understanding, let’s work through an example of how a changing electric field produces a magnetic field.

Example: Suppose we have an electric field given by (E = 4t\, V/m), where (t) is time in seconds. Find the magnetic field at (t = 2) seconds.

We can use Faraday’s Law to find the magnetic field. Taking the partial derivative of the electric field with respect to time:

 

\frac{\partial E}{\partial t} = 4\, V/m

The magnitude of the induced magnetic field will be 4 T (teslas) at (t = 2) seconds.

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