In Fig. 29-71, a long circular pipe with outside radius R=2.6 cm carries a (uniformly distribut

step 1 For part A of the problem, the field at the center of the pipe at a point C is due to the wire a

In Fig. 29-71, a long circular pipe with outside radius...

In Fig. $29-71,$ a long circular pipe with outside radius $R=2.6 \mathrm{~cm}$ carries a (uniformly distributed) current $i=8.00 \mathrm{~mA}$ into the page. A wire runs parallel to the pipe at a distance of $3.00 R$ from center to center. Find the (a) magnitude and (b) direction (into or out of the page) of the current in the wire such that the net magnetic field at point $P$ has the same magnitude as the net magnetic field at the center of the pipe but is in the opposite direction.

Key Concepts

Superposition Principle

The superposition principle states that the net magnetic field created by multiple currents is the vector sum of the magnetic fields generated by each current separately. This principle allows one to calculate the resultant magnetic field at a point by combining the effects of different current sources, even if they are produced by different geometries such as a pipe and a straight wire.

Ampère's Law

Ampère’s Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is used to compute the magnetic field generated by current distributions, especially in scenarios where the current is symmetrically distributed, such as in cylindrical conductors or pipes. The law simplifies the evaluation of the magnetic field by taking advantage of symmetry in the current configuration.

Magnetic Field of a Long Straight Conductor

A long straight conductor carrying a steady current produces a magnetic field that forms concentric circles around the wire. This magnetic field’s magnitude is given by an expression derived using Ampère's Law, which shows that the field decreases inversely with distance from the wire. This concept is crucial when determining the contribution of the magnetic field at a point not on the current-carrying conductor’s axis.

Uniform Current Distribution in Cylindrical Conductors

In cylindrical conductors or pipes where the current is uniformly distributed across the cross-section, the magnetic field both inside and outside the conductor can be determined by applying Ampère’s Law. The concept is important as it distinguishes the field behavior inside the conductor from that outside, allowing for accurate analysis of the field at specific points of interest.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the magnetic field generated by a current. By pointing the thumb of the right hand in the direction of current flow, the curled fingers indicate the direction of the magnetic field lines. This concept helps in predicting the direction (into or out of the page) of the magnetic field contributions from each current-carrying element.

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