# Electromagnetic Induction On A Charged Spherical Conducting Surface Problems And Solutions Pdf

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*Problems and Puzzles in Electric Fields pp Cite as. This chapter is the main part of the book, and contains the questions problems and puzzles and answers.*

- CBSE Syllabus For Class 12 Physics 2021 (Revised) | Download New Syllabus PDF
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- Electric Field Mcq Pdf

So far, we have generally been working with charges occupying a volume within an insulator. We now study what happens when free charges are placed on a conductor. Generally, in the presence of a generally external electric field, the free charge in a conductor redistributes and very quickly reaches electrostatic equilibrium. If an electric field is present inside a conductor, it exerts forces on the free electrons also called conduction electrons , which are electrons in the material that are not bound to an atom.

## CBSE Syllabus For Class 12 Physics 2021 (Revised) | Download New Syllabus PDF

Bueno 1. The study of electrostatic phenomena is the gateway to the physics described by Classical Electrodynamics. In this paper, we discuss in detail two methods based on the Uniqueness Theorem for solving electrostatic problems with azimuthal symmetry.

The first one is the electrostatic potential extension from the axis of symmetry to an arbitrary point. The other consists in the mutual mapping between two potentials through an inversion transformation.

We have prepared a list of six examples for which we calculate, completely or partially, the electrostatic potentials for different charge distributions using both methods.

The electric field lines are analyzed and presented graphically in all cases. Classical Electrodynamics CED , formulated at the end of the nineteenth century, is one of the greatest triumphs of science. It not only unified the already known electric and magnetic phenomena but also predicted the existence of electromagnetic waves and, being the first relativistic theory developed, CED served as a foundation for our current understanding of space and time.

Together with the gravitational interaction, the classical electromagnetic fields are responsible for all the physics we observe in our macroscopic daily life. In CED, the evolution of the electromagnetic e. At the same time, an electromagnetic field defined in space creates a Lorentz force on each charge q i given by. Therefore it is not difficult to see that the description of a system formed by electric charges and an e. In general, we have an endless loop: the electric charges create an e.

Only in simple systems , when we have control over the field configuration or of the charge distribution, there are analytical solutions. Fortunately, in macroscopic scales, a class of simple systems becomes very relevant — the electrostatic phenomena.

Textbooks of Basic Physics [ 1 2 — 3 ] and Classical Electromagnetic Field Theory [ 4 , 5 ] usually dedicate a substantial part of their text to the analysis of electrostatic physics. This article is devoted to the introduction and implementation of two powerful techniques described subtly in the references [ 6 , 7 ]. These methods are little explored in undergraduate courses and allow for the resolution sometimes only in a partial way of a wide range of electrostatic problems with azimuthal symmetry.

The first technique consists in the determination of the electrostatic potential by an explicit calculation done only on the axis of symmetry. The second technique is the inversion method , in which the potential on the outside of a sphere of radius R is mapped to the inside of it and vice versa, maintaining the boundary conditions on the sphere surface intact. In section 2, we have a review of the principal properties of the Poisson and Laplace equations that govern the electrostatic phenomena.

In section 3, the two methods are derived using the Uniqueness Theorem in the context of problems with azimuthal symmetry. Section 4 provides a series of applications that illustrate the advantages and limitations of the two methods.

Finally, we present our final considerations in section 5. In the absence of dynamics, there is no magnetic field and Maxwell's Eqs. The second equation implies that the electric field is conservative. Substituting this new form into the first equation of 3 we have the following result. The most important one is the Uniqueness Theoremml: it says that if Dirichlet or Neumann boundary conditions are given, respectively.

The proof of the theorem can found in several books on the subject [ 4 — 5 6 ]. In this article, we will deal only with Dirichlet boundary conditions. For a localized charge distribution, i.

The solution given by equation 6 has a problem at the practical level: even for simple charge configurations, it can lead to complicated integrals. We will discuss this point in section 3. An important fact is that the scalar potential has no minimum or maximum at the points where equation 7 is valid — Earnshaw's Theorem. As a consequence there is no distribution of static charges resulting in a stable configuration, i. One must take some caution in the theorem's proof, it is not uncommon the use of conceptually wrong arguments in this task, see reference [8] for a comprehensible discussion about the subject.

A correct and elegant demonstration is found on page 3 of [7]. We are interested in electrostatic situations with azimuthal symmetry. In the rest of the section, we shall introduce two methods that can help in this task. In this region, the potential is given simultaneously by Eqs. Let us assume that the integral 6 is difficult to solve at an arbitrary point, but it is simple along the z -axis of symmetry. A ring, a disk and a rod with uniform distributions of charges are examples of this type of situation, and we will explore them in the next section.

Performing the integration only on the axis of symmetry, we obtain the exact result. For these points, the Legendre Polynomials assume the values, see equation 2 Appendix A. This procedure can always be done allowing, a priori, the construction of two potentials for the price of one.

An interesting point to note is that by performing this operation twice, we return to the original potential, i. The main question here is when the Method of Inversion is useful. This occurs when we have boundary conditions exactly on the sphere of radius R. Here we will use the methods developed in the previous section to obtain the electrostatic potential in terms of power series for several examples.

The problems were chosen to illustrate the advantages and limitations of this approach. In this way, we will describe in all cases the electric field graphically using the Wolfram Mathematica software.

The calculation of the potential at an arbitrary point using equation 6 is not a trivial task. On the other hand, the integral only on the z -axis is a simple exercise illustrated in several basic physics books. The result is [1]. This exercise is the standard one to illustrate the method of extension around the axis of symmetry [6].

To extend this solution we must expand equation 14 in power series. Thus, as described in section 3. The complete solution is. Now the terms B l are all null and only the even parameters A l 's are non-zero, i.

The potential inside the sphere of radius R is. Continuity is a necessary condition, but the intersection of the two solutions being a sphere is a curious fact, since the problem is about a ring of charge. This is a consequence of the azimuthal symmetry. It is important to note that the calculation of the two solutions in a separate way was not necessary. One solution determines the other by the inversion method.

As one can see, in the inside of the spherical region of radius R the field lines move away from the plane of the ring; far outside of the ring, the lines approach a radial behavior, which is the expected result for a point charge. The two points represent the location of the ring on the plane; the dashed circle indicates the sphere of radius R. Our second example is "Saturn's ring". See figure 3a. Our approach will be via the method of images. The Uniqueness Theorem guarantees that if such a configuration exists, it creates the same potential as the original problem.

Using Eqs. The respective choices. The integral below provides the total induced charge Q i n d on the grounded conductor. As expected, this is the value of the total charge of the image ring. An important characteristic is that inside of the fictitious sphere of radius b the radial component of the electric field, i. As a consequence all field lines point inwards and terminate in the conducting surface. The field lines far away from the system have a radial aspect a point charge behavior.

The two black dots represent the intersection of the ring with the plane; the dashed circle indicates the sphere of radius b , and the full black circle is the grounded conductor of radius R. This exercise is solved respectively in sections 9. Here we will directly use the results derived in subsection 4. That is, the inversion method swapped the real ring and the image ring, keeping the sphere of radius R invariant, see figure 6b.

The final electrostatic potential is. The conducting sphere dashed circle is not affected by the operation. The two black dots represent the intersection of the ring with the plane, the dashed circle indicates the sphere of radius a , and the full black circle is the grounded conductor of radius R. The first step is to determine the potential in the z -axis of symmetry. Note that the zero-term cancels in the sum. By introducing the radial coordinate and the Legendre polynomials, the final form of the potential in this region is.

The results of section 3. In this case, Eqs. As expected the series 28 and 31 are different. This change of sign restores the negative sign in the second term, and following the previous steps we have equation 30 again. Assuming z positive and again with the help of equation 5 Appendix A we have. Introducing the Legendre polynomials we have the solution. Through direct calculation, it is possible to see that the expansion in the negative semi-axis provides the same result. The potential is continuous as it should be along the sphere of radius R ,.

It is easy to verify that. Far from the system, the electric field becomes radial, such as that of a point charge. The continuous black semicircle represents the hemisphere of radius R.

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The method of image charges also known as the method of images and method of mirror charges is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem see Dirichlet boundary conditions or Neumann boundary conditions. The validity of the method of image charges rests upon a corollary of the uniqueness theorem , which states that the electric potential in a volume V is uniquely determined if both the charge density throughout the region and the value of the electric potential on all boundaries are specified. Possessing knowledge of either the electric potential or the electric field and the corresponding boundary conditions we can swap the charge distribution we are considering for one with a configuration that is easier to analyze, so long as it satisfies Poisson's equation in the region of interest and assumes the correct values at the boundaries. This situation is equivalent to the original setup, and so the force on the real charge can now be calculated with Coulomb's law between two point charges.

There are a variety of methods to charge an object. One method is known as induction. In the induction process, a charged object is brought near but not touched to a neutral conducting object. The presence of a charged object near a neutral conductor will force or induce electrons within the conductor to move. The movement of electrons leaves an imbalance of charge on opposite sides of the neutral conductor.

## Questions and Answers

It is cheaper to buy co ee in New York at least according to the physics textbook that is. Write its SI unit and dimensions. If this sphere is connected to a ground through the wire as shown in fig 3 free electrons of the sphere at farther end flow to the ground. Sep 24 The reason is simple physics light behaves as a ray the speed of light in water is different than the speed of light in air and when light enters or leaves that medium it always bends in a Review Problems for Introductory Physics 1 May 20 Robert G.

It is a very important subject for various entrance exams for Engineering, Medical, or other higher education in the future. Class 12 Physics is the most important subject for the academic exam You should know that a thorough concept of Class 11 and Class 12 Physics to score better marks in the various entrance exam. With a better concept on the latest syllabus, you can secure good marks. You already know CBSE Class 12 syllabus will be updated almost every year to provide the students with the latest syllabus in real-time.

### Electric Field Mcq Pdf

We have over electronics and electrical engineering multiple choice questions MCQs and answers — with hints for each question. B what are the strength and direction of the electric field at the position indicated by the dot in the figure? The unit of solid angle is-steradian, 5. Electromagnetic Induction.

Bueno 1. The study of electrostatic phenomena is the gateway to the physics described by Classical Electrodynamics. In this paper, we discuss in detail two methods based on the Uniqueness Theorem for solving electrostatic problems with azimuthal symmetry.

#### Section Summary

As a result of doing this, the electroscope leaves 2. This post is going to cover Static electricity, electrostatic charge, and related topics with important questions and answers. These metal strips, called leaves, are attached to a central metal rod that has a metal sphere at the top. Notes for Photoelectric effects chapter of class 12 physics. What are the properties of electric charge? Place a radiation source inside and explain the effect it causes.

Electromagnetic Field Theory. Define a dipole. In some haunted locations, researchers have measured magnetic fields that are stronger than normal or which exhibit unusual fluctuations. Steady electrical currents 5. The electric field is produced by stationary charges, while the magnetic field is produced by moving charges, i.

The first results on the modelling of magnetic signals induced by transient equatorial ring currents in the magnetosphere with a timescale of the order of days and recorded at satellite altitudes are presented. The input of modelling consists of the X -component of the time-series recorded by the CHAMP vector magnetometer along individual night-time, mid-latitude satellite tracks. We have not considered the magnetic signals measured along day-time tracks and above polar regions because they are disturbed by signals with sources different from magnetospheric ring currents. The modelling procedure is divided into two parts. First, the X -components of satellite magnetic signals are processed by a two-step least-squares analysis. As a result, the X -data are represented in terms of series of Legendre polynomial derivatives.

As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf.

*Weiss, M. Long-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivity heterogeneities which are indicators of physiochemical variations found in the Earth's dynamic mantle. A prerequisite for mapping such heterogeneity is the ability to model accurately electromagnetic induction in a heterogeneous sphere.*

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Solution. The flux thru the Gaussian surface is the charge located inside the surface. spherical conducting shell with an inner radius a and an outer radius b. electric fields created by the induced charged on the inner and outer surfaces of.

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