In this article we are going to discuss 7 facts related to magnetic field and time.

**There is a relation between magnetic field and time or else it can be said that magnetic fields are used to vary with time. It means that the magnetic field is a time dependent vector quantity. If we focus on electromagnetic induction and the laws related to it we will be able to see that the magnetic field is affected by time.**

More specifically, we can conclude that the magnetic field is a function of time. Magnetic field is definitely time dependent. Surely we can say this using Faraday’s laws of electromagnetic induction and Lenz’s law.

**Is the magnetic field dependent on time?**

**We will take the help of Faraday’s laws of electromagnetic induction to show how the magnetic field depends on time. Basically current which is flowing through a closed loop,magnetic field,electric field all these quantities are interrelated to each other. To show how they are interrelated and how magnetic fields are dependent on time Faraday did three experiments.**

In the first experiment a loop of wire was pulled to the right of the magnetic field B and hence a flow of current occurred through the loop.

In the second experiment he kept the current carrying loop at rest and moved the magnet whose magnetic field is B towards lest. Current again flowed through the loop.

In the third experiment he kept both the magnet and the current carrying loop stationary but this time also there was a flow of current through the loop.

Now the result of the third case is a bit surprising. Why? Because we know that in the first case there is a motional emf(Ɛ) produced according to the **Faraday’s laws of electromagnetic induction whose value is,Ɛ = – dΦ/ dt**. Here when the loop moves basically a magnetic force is generated which in turn produces an emf. The second case is also the same as the first one. But the third case is different.

In the third case there is no movement of the magnet and the loop. So the question arises from where the magnetic force is generated as static charges can not generate magnetic force ,only moving charges can. Now the answer to this question is this force which generates the induced emf in this case is not a magnetic force. **Due to the presence of electric charges there is an electric field which in turn exerts a force on the loop and generates induced emf.**

Or else we can say that:

An electric field is induced by a changing magnetic field |

**Relation between electric and magnetic field**

Hence we can write that Ɛ = ∮ E. dl = – dΦ/ dt

⇒ ∮ E. dl = – d/dt(B.A) [ as Φ = B.A]

⇒ ∮ E. dl = -∫ ∂B/∂t . dA

This is the integral form of Faraday’s law. Here we can use Stokes’ theorem :

∇ x E = – ∂B/∂t |

**Relation between electric field and varying magnetic field**

From the equation that is written above we can say that the magnetic field is dependent on time.

**Relation between magnetic field and time**

**If a magnetic field is dependent on time, those types of fields are known as time varying magnetic fields. There must be a condition in this. In this case the magnetic field has to be a function of time. Otherwise it is not possible to find out the relation between these two quantities.**

There are some other types of magnetic fields too which are not dependent on time. For those kinds of magnetic fields we will not be able to find out the relation between them as these kinds of fields do not vary with time or it can be said that they are not functions of time.

From Faraday’s laws of electromagnetic induction we will be able to relate magnetic field and time. Induced emf Ɛ = – dΦ/ dt

Hence we can write that Ɛ = ∮ E. dl = – dΦ/ dt

⇒ ∮ E. dl = – d/dt(B.A) [ as Φ = B.A]

⇒ ∮ E. dl = -∫ ∂B/∂t . dA

This is the integral form of Faraday’s law. Here we can use Stokes’ theorem :

∇ x E = – ∂B/∂t |

**Relation between electric field and varying magnetic field**

**Why is a magnetic field dependent on time?**

**The change in the magnetic forces is the actual reason why does a magnetic field depends on time. These forces usually generate magnetic fields. magnetic field is a continuous flux whose value changes with respect to time.**

**Magnetic field and time graph**

**To draw a graph between magnetic field and time we have to take magnetic field B as a function of time t. We can take B as B(t) = B₀sin(π/2.t)ẑ**

We will take time t = 0 s, t = 1 s, t = 2 s, t = 3 s and t = 4 s to show how magnetic field varries with respect to time. After that we will plot these points on the graph to find out its nature. It is already cleared that graph will be sinusoidal in nature as B(t) is a sinusoidal function.

Now at t =0 second B = B₀ x sin 0 = 0

At t = 1 seconds B = B₀ x sin π/2 ẑ = B₀.ẑ

At t = 2 seconds B = B₀ x sin 2π/2 ẑ = 0

At t = 3 seconds B = B₀ x sin 3π/2 ẑ = – B₀.ẑ

At t = 4 seconds B = B₀ x sin 4π/2 ẑ = 0

Therefore the graph of magnetic field and time is:

Time( t in s) | Magnetic field( B ) |

t = 0 s | 0 |

t = 1 s | B₀.ẑ |

t = 2 s | 0 |

t = 3 s | – B₀.ẑ |

t = 4 s | 0 |

**Data for drawing the graph Magnetic field vs Time**

**How does a magnetic field vary with time?**

**We will not go to complex concepts of how a magnetic field varies with time. But we will try to explain it through simple words. So whenever the value of a magnetic field changes with respect to time it means that this is a time varying magnetic field. Now the magnetic field is basically a vector quantity that depends upon both space and time. We can write the magnetic field B as a function of (x,y,z) and time t.**

B = B (x,y,z,t) ≡ Bₓ ( x,y,z,t)

By (x,y,z,t)

Bz (x,y,z,t)

If ∂B/∂t = 0 , it is not a time varying magnetic field.

If ∂B/∂t ≠ 0 , it is a time varying magnetic field. So that type of magnetic field which is not constant with time but changes with time and gives a non zero time derivative is known as time varying magnetic field. If this non zero derivative is negative then the magnetic field is decreasing with time and if the value of the time derivative is positive then the value of magnetic field increases with time.

**Does the magnetic field decrease with time?**

**Here we will be going to describe why,how and when the magnetic field decreases with time. Already it has been clarified that when the magnetic field is a function of time then only we can say that magnetic fields vary with time i.e, the time varying magnetic fields. It means that it is not always possible that the magnetic field is decreasing with time,it is only possible when the magnetic field is a function of time. Let’s take an example of this. Say B(t) = B₀sin(π/2.t)ẑ**

We will take different values of time t to show that the magnetic field is decreasing with time.

Now at t =0 second B = B₀ x sin 0 = 0

At t = 1 seconds B = B₀ x sin π/2 ẑ = B₀.ẑ

At t = 2 seconds B = B₀ x sin 2π/2 ẑ = 0

At t = 3 seconds B = B₀ x sin 3π/2 ẑ = – B₀.ẑ

At t = 4 seconds B = B₀ x sin 4π/2 ẑ = 0

It is clear that at t = 3 seconds the value of magnetic field B is negative i.e, – B₀.ẑ. hence it can be said that at t = 3 seconds magnetic field will decrease for B(t) = B₀sin(π/2.t)ẑ.

**One Practice Problem**

**The equation of a magnetic field along the positive x axis is given by B(t) = 5cos****(****πt)î. What will be the value of induced emf after t = 4 seconds? Given value of the area is 5 m². Draw the graph of the given problem.**

**Answer :**

B(t) = 5cos(πt)î

We know, Induced emf Ɛ = – dΦ/ dt

➡ Ɛ = – d(B.A)/ dt

➡ Ɛ = – A.dB/ dt [ as A is constant = 5 m²]

➡ Ɛ = – 5 x d/dt [5cos(πt)î] = -5 x 5 x [- sin(πt)î ]

➡ Ɛ = 25 sin(πt)î

at t = 4 s ➡ Ɛ = 25 sin4πî = 0

**Graph**:

Time | Magnetic field |

0 s | 5î |

1 s | -5î |

2 s | 5î |

3 s | -5î |

4 s | 5î |

**Data for drawing the graph of magnetic field**

**Conclusion**

In this article we have described how a magnetic field varies with respect to time. It has also been shown that when the value of magnetic field decreases with time and the graph between magnetic field and time has also been given with proper explanation. At last there is one solved problem to show how we can apply these concepts in solving numericals.