Understanding Magnetic Fields from Perpendicular Coils

The question asks for the magnitude of the resultant magnetic field at the common center of two identical coils. These coils carry equal currents, and their planes are positioned at right angles (perpendicular) to each other. We are given that the magnetic field due to one coil alone at the center is \(B\).

Magnetic Field Direction at the Center

For a circular coil carrying current, the magnetic field at the center is directed along the axis of the coil. The direction is determined by the right-hand rule.

• Since the two coils are identical and carry equal currents, the magnitude of the magnetic field produced by each coil individually at their common center is the same, which is given as \(B\).
• The planes of the two coils are at right angles to each other. This means their axes are also at right angles to each other.
• Therefore, the magnetic field vector due to the first coil (\(\vec{B}_1\)) and the magnetic field vector due to the second coil (\(\vec{B}_2\)) at the common center will be perpendicular to each other. The magnitude of each vector is \(B\), i.e., \(|\vec{B}_1| = B\) and \(|\vec{B}_2| = B\).

Calculating the Resultant Magnetic Field

When two vector quantities are perpendicular to each other, their resultant magnitude can be found using the Pythagorean theorem, which is derived from vector addition.

Let the resultant magnetic field at the center be \(\vec{B}_{resultant}\). Since \(\vec{B}_1\) and \(\vec{B}_2\) are perpendicular, the magnitude of the resultant field is given by:

\begin{equation*} |\vec{B}_{resultant}| = \sqrt{|\vec{B}_1|^2 + |\vec{B}_2|^2} \end{equation*}

Substituting the magnitudes \(|\vec{B}_1| = B\) and \(|\vec{B}_2| = B\):

\begin{equation*} |\vec{B}_{resultant}| = \sqrt{B^2 + B^2} \end{equation*}

\begin{equation*} |\vec{B}_{resultant}| = \sqrt{2B^2} \end{equation*}

\begin{equation*} |\vec{B}_{resultant}| = B\sqrt{2} \end{equation*}

So, the magnitude of the resultant magnetic field at the center of the two perpendicular coils is \(\sqrt{2}B\).

Conclusion

The magnetic field due to one coil is \(B\). Due to the two identical coils with equal currents having their planes at right angles, the magnetic fields at the center are perpendicular vectors of magnitude \(B\). The resultant magnetic field magnitude is found by summing these perpendicular vectors.

The resultant magnetic field at the center is \(\sqrt{2}B\).