# How to Determine the Magnetic Field at the Center of a Partially Open Loop?

• slayer1337
In summary, the conversation discusses the determination of the magnetic field at a point P in the centre of a loop with a radius r, connected by two long straight sections with a gap subtending an angle of 60 degrees. The Biot-Savart law is used to calculate the magnetic field, and the equation for integrating dl/r^2 is discussed. The final solution involves using Ampere's law and the Biot-Savart law to calculate the field due to a continuous wire and a short section of wire connecting the loop.
slayer1337
1. An electric circuit consists of two long straight sections connecting a loop of radius r. The gap in the loop where the straight sections are connected subtends an angle of theta=60d, as shown in this figure (http://imgur.com/tseDHL6). Determine the magnetic field B at the point P in the centre of the loop, if a current I is flowing through the circuit.

See the diagram here - http://imgur.com/tseDHL6

2. Biot-Savart Law
3. The Attempt at a Solution - http://imgur.com/QFrlSn7

I have found the equation relating to the full loop but don't know how to do it for the missing segment.

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What's wrong with integrating ##d l\over r^2## ?

You need to use the Biot-Savart law:

Ma is right, but the next step is to convert to integrating ##d l\over r^2## because there is only one component left over; the others cancel.

$\int rd θ/r^2/$

Consider finding the field as though the circular section wasn't there, but the wire was continuous from -infinity to +infinity: B1. Can use ampere's law.

Then, use Biot-Savart to compute the field due to a short section of wire connecting the loose ends of the loop: B2.

Then, use Biot-Savart to compute the field due to the loop: B3.

B = B1 - B2 + B3.

## 1. What is the Biot-Savart law?

The Biot-Savart law is a fundamental law in electrodynamics that describes the magnetic field generated by a steady current. It states that the magnetic field at a given point is directly proportional to the magnitude of the current, the length of the current-carrying wire, and the sine of the angle between the wire and the point in question.

## 2. How is the Biot-Savart law different from Ampere's law?

While both laws deal with the magnetic field, the Biot-Savart law is used to calculate the magnetic field due to a steady current, whereas Ampere's law is used to calculate the magnetic field due to a steady current enclosed by a closed loop.

## 3. What is the mathematical representation of Ampere's law?

Ampere's law is represented as ∮B•dl = μ0I, where ∮B•dl represents the line integral of the magnetic field around a closed loop, μ0 is the permeability of free space, and I is the current enclosed by the loop.

## 4. Can the Biot-Savart law and Ampere's law be used to calculate the magnetic field in all situations?

No, both laws have limitations and can only be used to calculate the magnetic field in certain situations. For example, the Biot-Savart law is only applicable to steady currents, while Ampere's law is only applicable to steady currents enclosed by a closed loop.

## 5. How are the Biot-Savart law and Ampere’s law related?

The Biot-Savart law is one of the fundamental principles used to derive Ampere's law. In fact, Ampere's law can be seen as a generalization of the Biot-Savart law, as it includes the effects of multiple currents and their directions on the magnetic field.

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