Informative line

### Direction Of Magnetic Field

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.

# Cross Product of Two Vectors

• 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,

$$\vec A×\vec B=-(\vec B×\vec A)$$

#### The direction of $$\vec A×\vec B$$ will be

A $$-\hat k$$

B $$\hat k$$

C $$\hat j$$

D $$-\hat j$$

×

According to right hand thumb rule, the direction of $$\vec A×\vec B$$ will be perpendicular to the plane of  $$(\vec A×\vec B)$$.

Since, $$(\vec A×\vec B)$$ lie in x-y plane. So, the direction of $$(\vec A×\vec B)$$ will be in the direction of unit vector $$\hat k$$.

### The direction of $$\vec A×\vec B$$ will be

A

$$-\hat k$$

.

B

$$\hat k$$

C

$$\hat j$$

D

$$-\hat j$$

Option B is Correct

# Direction of Magnetic Field (B) at a Point due to Point Charge

• 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 positive charge is moving in +y direction with a velocity $$\vec v$$. The direction of magnetic field at point P whose z-coordinate is zero, as shown in figure, will be along unit vector -

A $$\hat i$$

B $$-\hat k$$

C $$\hat k$$

D $$\hat j$$

×

Since, $$\vec v$$ and $$\vec r$$ lies in x-y plane. So, direction of $$(\vec v×\vec r)$$ will be along $$\hat k$$.

Hence, the direction of magnetic field will be perpendicular to plane of $$(\vec v\;and\;\vec r)$$.

So, the direction of magnetic field is along $$\hat k$$.

### A positive charge is moving in +y direction with a velocity $$\vec v$$. The direction of magnetic field at point P whose z-coordinate is zero, as shown in figure, will be along unit vector -

A

$$\hat i$$

.

B

$$-\hat k$$

C

$$\hat k$$

D

$$\hat j$$

Option C is Correct

# Magnetic Field due to Small Element of Current Carrying Wire

• 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.

#### A current carrying wire is shown in figure. The direction of magnetic field at point A due to small element PQ as shown in figure will be - [Assume the element to be very small than the distance of the point from the element ]

A Upward

B Downward

C Inwards

D Outwards

×

Considering a small element PQ of wire to calculate electric field at P.

Since the wire is very small it can be considered as a straight line.

According to right hand thumb rule,

The direction of magnetic field at P will be in outward direction.

### A current carrying wire is shown in figure. The direction of magnetic field at point A due to small element PQ as shown in figure will be - [Assume the element to be very small than the distance of the point from the element ]

A

Upward

.

B

Downward

C

Inwards

D

Outwards

Option D is Correct

# Magnetic Field at the Axis of a Current Carrying Loop due to Small Section of Wire

• 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 rectangular wire loop carrying a current in the direction  as shown in figure is placed in x-z plane. The direction of magnetic field at point P due to side CD of the rectangular loop will be along unit vector

A $$\hat i$$

B $$\hat j$$

C $$-\hat k$$

D $$-\hat j$$

×

By applying right hand thumb rule, the direction of magnetic field will be along $$-\hat j$$.

Front view of arm CD

### A rectangular wire loop carrying a current in the direction  as shown in figure is placed in x-z plane. The direction of magnetic field at point P due to side CD of the rectangular loop will be along unit vector

A

$$\hat i$$

.

B

$$\hat j$$

C

$$-\hat k$$

D

$$-\hat j$$

Option D is Correct

# Direction of Magnetic Field for a Negative Charge

• 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.

#### Find the direction of magnetic field at 0 if a negative charge is moving upward from point P (4, 5).

A $$\hat i$$

B $$-\hat k$$

C $$\hat k$$

D $$\hat j$$

×

Direction of magnetic field when positive charge is placed

Since, $$\vec r$$ and $$\vec v$$ both are in x-y plane. So, direction of magnetic field will be perpendicular to the plane of $$\vec v \,\,\,$$and$$\,\,\,\vec r$$ for positive charge (according to right hand thumb rule).

So, direction of magnetic field for positive charge = $$\hat k$$

For negative charge, direction of magnetic field will be opposite to that of positive charge.

Direction of magnetic field at O = Direction of $$(\vec r×\vec v)$$ = –direction of $$(\vec v×\vec r)$$

$$-\hat k$$

### Find the direction of magnetic field at 0 if a negative charge is moving upward from point P (4, 5).

A

$$\hat i$$

.

B

$$-\hat k$$

C

$$\hat k$$

D

$$\hat j$$

Option B is Correct

# Direction of Magnetic Field due to Current Carrying Wire

• Consider a current carrying wire as shown in figure.

• 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.

#### In all the options, a current carrying wire and a point (P) are shown. Choose the option in which the direction of magnetic field is not outwards.

A

B

C

D

×

Direction of magnetic field $$(\vec B)$$ can be determined by right hand thumb rule.

In option A

Since, fingers curl outward, the direction of magnetic field is outward.

In option B

Since, fingers curl outward, the direction of magnetic field is outward.

In option C

Since, fingers curl outward, the direction of magnetic field is outward.

In option D

Since, fingers curl inward, the direction of magnetic field will be inward.

### In all the options, a current carrying wire and a point (P) are shown. Choose the option in which the direction of magnetic field is not outwards.

A
B
C
D

Option D is Correct