Magnetostatics

Concept: Magnetic field due to a current element is given by the Biot–Savart law:

d%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B%5Cmu_0%7D%7B4%5Cpi%7D%20%5Cfrac%7BI%20%5C%2C%20d%5Cmathbf%7Bl%7D%20%5Ctimes%20%5Cmathbf%7B%5Chat%7Br%7D%7D%7D%7Br%5E2%7D

Vector form:

d%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B%5Cmu_0%7D%7B4%5Cpi%7D%20%5Cfrac%7BI%20%5C%2C%20(d%5Cmathbf%7Bl%7D%20%5Ctimes%20%5Cmathbf%7Br%7D)%7D%7Br%5E3%7D

Step 1: Given data.

Length of segment:

dl%20%3D%201%20%5C%2C%20%5Ctext%7Bcm%7D%20%3D%2010%5E%7B-2%7D%20%5C%2C%20%5Ctext%7Bm%7D

Current:

I%20%3D%2010%20%5C%2C%20%5Ctext%7BA%7D

Segment along +x direction:

d%5Cmathbf%7Bl%7D%20%3D%2010%5E%7B-2%7D%20%5C%2C%20%5Chat%7Bi%7D

Field point:

(1%2C1%2C0)

Position vector:

%5Cmathbf%7Br%7D%20%3D%20%5Chat%7Bi%7D%20%2B%20%5Chat%7Bj%7D

r%20%3D%20%5Csqrt%7B1%5E2%20%2B%201%5E2%7D%20%3D%20%5Csqrt%7B2%7D

Step 2: Compute cross product.

d%5Cmathbf%7Bl%7D%20%5Ctimes%20%5Cmathbf%7Br%7D%0A%3D%20(10%5E%7B-2%7D%20%5Chat%7Bi%7D)%20%5Ctimes%20(%5Chat%7Bi%7D%20%2B%20%5Chat%7Bj%7D)

Using cross products:

%5Chat%7Bi%7D%20%5Ctimes%20%5Chat%7Bi%7D%20%3D%200%2C%20%5Cquad%20%5Chat%7Bi%7D%20%5Ctimes%20%5Chat%7Bj%7D%20%3D%20%5Chat%7Bk%7D

d%5Cmathbf%7Bl%7D%20%5Ctimes%20%5Cmathbf%7Br%7D%20%3D%2010%5E%7B-2%7D%20%5Chat%7Bk%7D

Step 3: Apply Biot–Savart law.

d%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B%5Cmu_0%7D%7B4%5Cpi%7D%20%5Cfrac%7BI%20(10%5E%7B-2%7D%20%5Chat%7Bk%7D)%7D%7B(%5Csqrt%7B2%7D)%5E3%7D

(%5Csqrt%7B2%7D)%5E3%20%3D%202%5Csqrt%7B2%7D

d%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B10%5E%7B-7%7D%20%5Ctimes%2010%20%5Ctimes%2010%5E%7B-2%7D%7D%7B2%5Csqrt%7B2%7D%7D%20%5Chat%7Bk%7D

Step 4: Simplify.

10%5E%7B-7%7D%20%5Ctimes%2010%20%5Ctimes%2010%5E%7B-2%7D%20%3D%2010%5E%7B-8%7D

%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B10%5E%7B-8%7D%7D%7B2%5Csqrt%7B2%7D%7D%20%5Chat%7Bk%7D%0A%3D%20%5Cfrac%7B10%5E%7B-8%7D%7D%7B2.828%7D%20%5Chat%7Bk%7D

%5Cmathbf%7BB%7D%20%5Capprox%203.5%20%5Ctimes%2010%5E%7B-9%7D%20%5C%2C%20%5Chat%7Bk%7D%20%5C%2C%20%5Ctext%7BT%7D

Final Answers:

Vector expression (Biot–Savart law):

d%5Cmathbf%7BB%7D%20%3D%20%5Cfrac%7B%5Cmu_0%7D%7B4%5Cpi%7D%20%5Cfrac%7BI%20(d%5Cmathbf%7Bl%7D%20%5Ctimes%20%5Cmathbf%7Br%7D)%7D%7Br%5E3%7D

Magnetic field at

(1%2C1%2C0)

%5Cmathbf%7BB%7D%20%5Capprox%203.5%20%5Ctimes%2010%5E%7B-9%7D%20%5C%2C%20%5Chat%7Bk%7D%20%5C%2C%20%5Ctext%7BT%7D

Chapter Questions
1 MCQs