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Chapter 22: Problem 13

Why do electric field lines never cross?

Short Answer

Expert verified
Answer: Electric field lines never cross because the electric field is unique at any point in space. Crossing field lines would imply that there is more than one value of electric field intensity at a given point, violating the fundamental laws of physics. This would lead to ambiguous conclusions about the direction and magnitude of the force acting on a charged particle at that location.

Step by step solution

01

Understanding the electric field

The electric field is a vector field, which means that at every point in space, there is an electric field vector associated with it. This vector represents the force experienced by a unit positive charge if it were placed at that point. That means the direction of the vector represents the direction of the force on the charge, and its magnitude represents the strength of the force.
02

Defining electric field lines

Electric field lines are a visual representation of how electric charge would influence the space around it. The lines are drawn so that the tangent to the line at any point represents the direction of the electric field at that point. The spacing of the lines is also significant; denser field lines indicate a stronger electric field, whereas sparser lines indicate a weaker field.
03

Explaining the uniqueness of electric field direction

The main reason electric field lines never cross is because of the uniqueness of the electric field direction at any point in space. If electric field lines were to cross, there would be two different electric field vectors (meaning two different directions) at that location. This would mean that a charged particle placed at the point of intersection would experience two different simultaneous forces, which is contradictory to the concept of electric field - the field acts with one unique force on the charge at that given point.
04

Visualizing the field lines of multiple charges

When we have more than one charge present, the total electric field at a point is the vector sum of the individual electric fields created by each charge. Although the individual field lines may appear to come close to each other, they essentially reshape themselves in response to the total electric field vector, ensuring that the lines never actually cross.
05

Understanding the consequences of crossing field lines

If electric field lines were to cross, it would imply that there is more than one value of electric field intensity at a given point in space, violating the fundamental laws of physics that state the electric field is unique at any point in space. This would lead to ambiguous conclusions about the direction and magnitude of the force acting on a charged particle at that location. In conclusion, electric field lines never cross because the electric field is unique at any point in space, and the lines serve as a visual representation of the direction and magnitude of the force on a charged particle at any given location.

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Most popular questions from this chapter

A conducting solid sphere of radius \(20.0 \mathrm{~cm}\) is located with its center at the origin of a three-dimensional coordinate system. A charge of \(0.271 \mathrm{nC}\) is placed on the sphere. a) What is the magnitude of the electric field at point \((x, y, z)=\) \((23.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?\) b) What is the angle of this electric field with the \(x\) -axis at this point? c) What is the magnitude of the electric field at point \((x, y, z)=\) \((4.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?\)

Suppose you have a large spherical balloon and you are able to measure the component \(E_{n}\) of the electric field normal to its surface. If you sum \(E_{n} d A\) over the whole surface area of the balloon and obtain a magnitude of \(10 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\) what is the electric charge enclosed by the balloon?

A solid nonconducting sphere has a volume charge distribution given by \(\rho(r)=(\beta / r) \sin (\pi r / 2 R) .\) Find the total charge contained in the spherical volume and the electric field in the regions \(rR\). Show that the two expressions for the electric field equal each other at \(r=R\).

Two infinite, uniformly charged, flat nonconducting surfaces are mutually perpendicular. One of the surfaces has a charge distribution of \(+30.0 \mathrm{pC} / \mathrm{m}^{2}\), and the other has a charge distribution of \(-40.0 \mathrm{pC} / \mathrm{m}^{2}\). What is the magnitude of the electric field at any point not on either surface?

A single positive point charge, \(q,\) is at one corner of a cube with sides of length \(L\), as shown in the figure. The net electric flux through the three net electric flux through the three adjacent sides is zero. The net electric flux through each of the other three sides is a) \(q / 3 \epsilon_{0}\). b) \(q / 6 \epsilon_{0}\). c) \(q / 24 \epsilon_{0}\). d) \(q / 8 \epsilon_{0}\).
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