Learn magnetic field definition with right hand rule magnetic field as cross product of two vectors, practice to calculate direction of magnetic field around a wire & due to current carrying wire and magnetic field of a point charge and a moving charge.

- Consider two vectors \(\vec A\) and \(\vec B\) making an angle \(\theta\) with each other.

- The cross product of \(\vec A\) and \(\vec B\) is defined as

\(\vec A×\vec B=|\vec A|\,|\vec B|sin\theta \,\,\hat n\)

where \(\hat n\) is the unit vector in the direction of \(\vec A×\vec B\).

- Cross product is also known as vector product as cross product of two vector quantities gives a vector quantity.

**Direction of \((\vec A×\vec B)\)**

- The direction of \((\vec A×\vec B)\) is specified by using the right hand thumb rule which states,

Curl the fingers of your right hand from \(\vec A\) towards \(\vec B\) with your thumb extending outwards and perpendicular to the plane of \(\vec A\) and \(\vec B\) as shown in figure.

- The direction of \((\vec A×\vec B)\) is now shown by your thumb.

Fingers curl from \(\vec A\) towards \(\vec B\) and thumb pointing towards direction of \((\vec A×\vec B)\).

**Cross product of standard unit vectors :**

Direction of cross product of two vector quantities can be understood using standard unit vectors.

Three unit vectors in 3-coordinate system, is shown in figure.

- While moving in the direction of the arrows shown in figure, cross product of any two unit vectors gives the third unit vector.

\(\hat i× \hat j=\hat k\)

\(\hat j× \hat k=\hat i\)

\(\hat k× \hat i=\hat j\)

- While moving in the opposite direction of the arrows shown in figure, cross product of any two unit vector gives negative of the third unit vector.
- \(\hat k× \hat j=-\hat i\)
\(\hat j× \hat i=-\hat k\)

\(\hat i× \hat k=-\hat j\)

- This can also be understood by,

\(\vec A×\vec B=-(\vec B×\vec A)\)

- Consider a positive point charge moving with some velocity \(\vec v\), as shown in figure.

- The direction of magnetic field at P will be determined using right hand thumb rule.
- Place your right hand around the line in such a way that your fingers sweep \(\vec v\) into \(\vec r\) through the smaller angle between them. The outstretched thumb will point in the direction of magnetic field \((\vec B)\).

OR

- Point the right thumb in the direction of \(\vec v\). The magnetic field vector \(\vec B\) is perpendicular to the plane of \(\vec r\) and \(\vec v\), pointing in the direction in which fingers curl.

**Case: Point charge going inside the page **

- To calculate magnetic field for a point charge going inside the page, use right hand thumb rule.

- Direct your right hand thumb inside the page. Now, curl your fingers, it will give the direction of magnetic field.

A \(\hat i\)

B \(-\hat k\)

C \(\hat k\)

D \(\hat j\)

- Consider a long, current carrying wire, as shown in figure.
- To calculate magnetic field due to small element of wire, a small element AB is selected.

- To determine the direction of magnetic field at a point P which is far away from the element, consider the element of any arbitrary shape to be straight line.

- The direction of magnetic field at point P can be specified by assuming the wire to be straight line.

- Applying right hand thumb rule, the direction of magnetic field at P by this wire is outward.

**Example :**

Direction of magnetic field at P due to small element AB of a current carrying wire loop can be specified by assuming the element as a straight wire.

- Considering a small element AB of wire.

- By applying right hand thumb rule, it is concluded that direction of magnetic field is inward.

- Consider a wire loop placed in y-z plane such that it is symmetrical along x-axis.

- Consider an element ds on this wire loop and a point P on the x-axis as shown in figure.

- Magnetic field at point P due to small element ds can be specified by considering the wire as straight line.
- By seeing it from front the direction of current in ds is outward.

- By applying right hand thumb rule, the direction of magnetic field at point P is shown in figure.

A \(\hat i\)

B \(\hat j\)

C \(-\hat k\)

D \(-\hat j\)

- Consider a negative charge going inside a page.

- To determine the direction of magnetic field at point P as shown in figure, following steps are to be followed.

**Step 1: **Find the direction of magnetic field considering negative charge as a positive charge.

**Step 2: **Opposite the direction of magnetic field.

A \(\hat i\)

B \(-\hat k\)

C \(\hat k\)

D \(\hat j\)

- Consider a current carrying wire as shown in figure.

- To find the direction of magnetic field about this wire, use right hand thumb rule.
- According to right hand thumb rule, if the right hand thumb is placed in the direction of the current and fingers are curled around the wire representing a circle, then the finger point in the direction of magnetic field around the wire.

**Example - **

- Consider a current carrying wire in which the current is flowing inward.