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Welcome to our Physics lesson on **How to Find the Induced Magnetic Field?**, this is the fourth lesson of our suite of physics lessons covering the topic of **Maxwell Equations**, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

In tutorial 16.2 "Magnetic Field Produced by Electric Currents" we explained that the direction of magnetic field produced by a current-carrying wire is found by using the right-hand rule, i.e. we grasp the wire with the right hand outstretching the thumb in the direction of current. In this case, the other four curled fingers show the direction of magnetic field.

We can apply the same rule for the magnetic field produced by the displacement current as well. In this case, we consider the cylinder formed by the capacitor plates and the space between them as a cylindrical conducting wire of radius R. Hence, applying the known equations derived in the tutorial 16.2, we obtain for the magnetic field produced by the displacement current inside the capacitor plates, at a distance r from the central axis (r < R),

B = *μ*_{0} ∙ i_{d}*/**2π ∙ R*^{2} ∙ r

and for the magnetic field outside the plates (at r > R from the central axis),

B = *μ*_{0} ∙ i_{d}*/**2π ∙ r*

A circular parallel-plate capacitor with plate radius R = 4 cm is being charged by a current I = 5A.

What is the magnitude of __∮__B ∙ dL at a distance of R/7 from the central point of capacitor plates?

What is the magnitude of magnetic field at this distance?

First, let's calculate the magnitude of the displacement current in the given section between the capacitor plates. Since the magnitude of the displacement current I_{d} represents a part of the real current I which charges the capacitor, the ratio between the area encircled by the loop of radius r to the total area between the plates is equal to the ratio of the two abovementioned currents. Thus, we have

Hence, the displacement current at the given position is

i_{d} = I ∙ *r*^{2}*/**R*^{2}

= I ∙*R**/**7**/**R*^{2}

=*I**/**49*

=*(5 A)**/**49*

= 0.102 A

= I ∙

=

=

= 0.102 A

Therefore, since at the given distance

we obtain for the value of integral

= 1.28 × 10

Since r = R/5 represents a location inside the plates, we have for the magnetic field B:

B = *μ*_{0} ∙ i_{d}*/**2π ∙ R*^{2} ∙ r

=*(4π × 10*^{-7} *N**/**A*^{2} ) ∙ (0.102 A)*/**2π ∙ (4 × 10*^{-2} m)^{2} ∙ *4 × 10*^{-2} m*/**5*

= 1.02 × 10^{-3} T

=

= 1.02 × 10

You have reached the end of Physics lesson **16.18.4 How to Find the Induced Magnetic Field?**. There are 5 lessons in this physics tutorial covering **Maxwell Equations**, you can access all the lessons from this tutorial below.

Enjoy the "How to Find the Induced Magnetic Field?" physics lesson? People who liked the "Maxwell Equations lesson found the following resources useful:

- Induced Magnetic Field Feedback. Helps other - Leave a rating for this induced magnetic field (see below)
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- Check your calculations for Magnetism questions with our excellent Magnetism calculators which contain full equations and calculations clearly displayed line by line. See the Magnetism Calculators by iCalculator™ below.
- Continuing learning magnetism - read our next physics tutorial: Introduction to Magnetism

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