# Magnetic Field Line Integral Problem

• cjhseeker
In summary, the line integral of B between points i and f can be calculated using Ampere's Law, which states that the integral of B dotted with dl is equal to the enclosed current multiplied by the constant μ0. Since the magnetic field is perpendicular to the surface on the straight segments, their contribution to the integral is zero. However, on the semicircle, the dl component is parallel to the B field, resulting in an integral of B multiplied by pi times the radius (0.01 meters). The right side of the equation should be 2A multiplied by μ0. However, it is unclear whether the problem is asking for the value of the magnetic field or the integral itself, and there is some confusion about the value
cjhseeker

## Homework Statement

What is the line integral of B between points i and f in the figure?

## Homework Equations

Ampere's Law: ∫B∙dl = Ienclosed * μ0
note: the integral on the left is a line integral.

## The Attempt at a Solution

I applied Ampere's Law. I know that the contribution of B to the line integral on the straight segments to the left and right of the semicircle is zero since the magnetic field is perpendicular to the surface (line) at these segments. The semicircle, however, has a dl component that is parallel to the B field at all points along the semicircle. Therefore the integral should be B*pi*r (where r = 0.01 meters) and the right side of the equation should be 2A * μ0. I am confused as to whether the problem is asking for the value of the magnetic field, or whether I am supposed to simply give the value of ∫B∙dl.

They want the integral.

pam said:
They want the integral.

In that case, the integral is equal to the current enclosed by the surface, multiplied by the constant (mu). Is the value of the current enclosed 2A or 1A? This is a bit unclear to me due to the nature of the Amperian surface. Thank you for helping.

The Amperian loop must be closed. Take it as a full circle for Ampere's law. Then, because of the symmetry, the integral on the half circle in the picture is half the full integral.

## 1. What is a magnetic field line integral problem?

A magnetic field line integral problem involves calculating the work done by a magnetic field on a charged particle as it moves along a closed path.

## 2. How do you calculate a magnetic field line integral?

To calculate a magnetic field line integral, you need to integrate the dot product of the magnetic field vector and the displacement vector along the closed path. This can be done using mathematical equations and principles of vector calculus.

## 3. What is the significance of solving a magnetic field line integral problem?

Solving a magnetic field line integral problem can help in understanding the behavior of charged particles in magnetic fields and can also be used to calculate the energy and work done in magnetic systems.

## 4. What factors affect the value of a magnetic field line integral?

The value of a magnetic field line integral is affected by the strength and direction of the magnetic field, the charge and velocity of the particle, and the shape and size of the closed path.

## 5. What are some real-world applications of magnetic field line integrals?

Magnetic field line integrals have various applications in fields such as electromagnetism, electrical engineering, and astrophysics. They are used in designing magnetic systems, understanding the behavior of particles in magnetic fields, and studying celestial bodies such as planets and stars.

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