Physics Lesson 16.18.4 - How to Find the Induced Magnetic Field?

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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.

How to Find the Induced Magnetic Field?

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 = μ0 ∙ id/2π ∙ R2 ∙ r

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

B = μ0 ∙ id/2π ∙ r

Example 3

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?

Solution 3

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 Id 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

id/I = π ∙ r2/π ∙ R2

Hence, the displacement current at the given position is

id = I ∙ r2/R2
= I ∙ R/7/R2
= I/49
= (5 A)/49
= 0.102 A

Therefore, since at the given distance

BdL = μ0 ∙ id

we obtain for the value of integral

BdL = (4π × 10-7 N/A2 ) ∙ (0.102 A)
= 1.28 × 10-7 N/A

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

B = μ0 ∙ id/2π ∙ R2 ∙ r
= (4π × 10-7 N/A2 ) ∙ (0.102 A)/2π ∙ (4 × 10-2 m)24 × 10-2 m/5
= 1.02 × 10-3 T

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.

More Maxwell Equations Lessons and Learning Resources

Magnetism Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
16.18Maxwell Equations
Lesson IDPhysics Lesson TitleLessonVideo
16.18.1Gauss Law for Magnetic Field
16.18.2Induced Magnetic Fields
16.18.3Displacement Current
16.18.4How to Find the Induced Magnetic Field?
16.18.5Maxwell Equations Explained

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