What Produces a Magnetic Field? — Question 4

Question 4

A local magnetic null can coexist with several nonzero sources

At point P, two fixed sources produce $3\hat x$ and $4\hat y$ microtesla. A control coil produces $\kappa I\hat u$, where $\kappa=2.00$ microtesla per ampere and $\hat u=-(3\hat x+4\hat y)/5$.

1. Find the signed current that makes the total field at P zero.

2. If this current is increased by .100 A, find the residual field vector and magnitude. Does a null prove the sources are absent?

Question 4 - Solution

A local magnetic null can coexist with several nonzero sources

(a) Method and calculation

$$\vec B_P=(3\hat x+4\hat y)+2I[-(3\hat x+4\hat y)/5]\quad\mathrm{\mu T}.$$$$1-2I/5=0\Rightarrow I=2.50\,\mathrm A.$$

(b) Interpretation and further calculation

$$\delta\vec B=\kappa\delta I\hat u=(-.120\hat x-.160\hat y)\,\mathrm{\mu T}.$$$$|\delta\vec B|=.200\,\mathrm{\mu T};\quad\vec B_P=0\ \not\Rightarrow\ \vec B_1=\vec B_2=0.$$

Physical check

A null is a vector cancellation at the specified point, not a statement that the entire region is field-free.